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THE

HINDU-ARABIC NUMERALS

DAVID EUGENE SMITH

AND

LOUIS CHARLES KARPINSIvI

BOSTON AND LONDON
GINN AND COMPANY, PUBLISHERS
1911

rCLCo,

HI3TOKICAL

MEDICAL

R k*

IK be gt be nit um jlrcgs

GINN AND COMPANY • PRO-
PRIETORS • BOSTON • U.S.A.

PREFACE

So familiar are we with the numerals that bear the
misleading name of Arabic, and so extensive is their use
in Europe and the Americas, that it is difficult for us to
realize that their general acceptance in the transactions
of commerce is a matter of only the last four centuries,
and that they are unknown to a very large part of the
human race to-day. It seems strange that such a labor-
saving device should have struggled for nearly a thou-
sand years after its system of place value was perfected
before it replaced such crude notations as the one that
the Roman conqueror made substantially universal in
Europe. Such, however, is the case, and there is prob-
ably no one who has not at least some slight passing
interest in the story of this struggle. To the “mathema-
tician and the student of civilization the interest is gen-
erally a deep one ; to the teacher of the elements of
knowledge the interest may be less marked, but never-
theless it is real ; and even the business man who makes
daily use of the curious symbols by which we express
the numbers of commerce, cannot fail to have some
appreciation for the story of the rise and progress of
these tools of his trade.

This story has often been told in part, but it is a long
tune since any effort has been made to bring together
the fragmentary narrations and to set forth the gen-
eral problem of the origin and development of these

iii

THE HINDTJ-AR ABIC NUMERALS

numerals. In this little work we have attempted to state
the history of these forms in small compass, to place
before the student materials for the investigation of the
problems involved, and to express as clearly as possible
the results of the labors of scholars who have studied
the subject in different parts of the world. We have
had no theory to exploit, for the history of mathematics
has seen too much of this tendency already, but as far
as possible we have weighed the testimony and have set
forth what seem to be the reasonable conclusions from
the evidence at hand.

To facilitate the work of students an index has been
prepared which we hope may be serviceable. In this the
names of authors appear only when some use has been
made of then’ opinions or when then works are first
mentioned in full in a footnote.

If this work shall show more clearly the value of our
number system, and shall make the study of mathematics
seem more real to the teacher and student, and shall offer
material for interesting some pupil more fully in his work
with numbers, the authors will feel that the considerable
labor involved in its preparation has not been in vain.

We desire to acknowledge our especial indebtedness
to Professor Alexander Ziwet for reading all the proof,
as well as for the digest of a Russian work, to Professor
Clarence L. Meader for Sanskrit transliterations, and to
Mr. Steven T. Byington for Arabic transliterations and
the scheme of pronunciation of Oriental names, and also
our indebtedness to other scholars in Oriental learning
for information.

DAVID EUGENE SMITH
LOUIS CHARLES KARPIN SKI

CONTENTS

CHAPTER PAGE

PRONUNCIATION OF ORIENTAL NAMES vi

I. EARLY IDEAS OF THEIR ORIGIN 1

* II. EARLY HINDU FORMS WITH NO PLACE VALUE . . 12

III. LATER HINDU FORMS, WITH A PLACE VALUE ... 38

IV. THE SYMBOL ZERO : 51

V. THE QUESTION OF THE INTRODUCTION OF THE

NUMERALS INTO EUROPE BY BOETHIUS .... 63

VI. THE DEVELOPMENT OF THE NUMERALS AMONG THE

ARABS 91

VII. THE DEFINITE INTRODUCTION OF THE NUMERALS

INTO EUROPE 99

VIII. THE SPREAD OF THE NUMERALS IN EUROPE ... 128

•

INDEX 153

PRONUNCIATION OP ORIENTAL NAMES

(S) - in Sanskrit names and words ; (A) = in Arabic names and words.

b, d, f, g, h, j, 1, m, n, p, sh (A), t,
th (A), v, w, x, z, as in English.

a, (S) like u in but : thus pandit,
pronounced pundit. (A) like a in
ask or in man. a, as in father.

c, (S) like cli in church (Italian c in
cento).

d, n, s, t, (S) d, 71, sh, t, made with
the tip of the tongue turned up
and hack into the dome of the
palate, d, s, t, z, (A) d, s, t, z,
made with the tongue spread so
that the sounds are produced
largely against the side teeth.
Europeans commonly pronounce
d, n, s, t, z, both (S) and (A), as
simple d, n, sh (S) or s (A), t, z.
d (A) , like th in this.

e, (S) as in they. (A) as in bed.

g, (A) a voiced consonant formed
below the vocal cords; its sound
is compared by some to a g, by
others to a guttural r ; in Arabic
words adopted into English it is
represented by gh (e.g. ghoul),
less often r (e.g. razzia i).

h preceded by b, c, t, t, etc. does
not form a single sound with these
letters, but is a more or less dis-
tinct h sound following them ; cf .
the sounds in abhor, boathook,
etc., or, more accurately for (S),
the “ bhoys ” etc. of Irish brogue,
h (A) retains its consonant sound
at the end of a word, h, (A) an
unvoiced consonant formed below
the vocal cords ; its sound is some-
times compared to German hard
ch, and may be represented by an
li as strong as possible. In Arabic
words adopted into English it is
represented by h, e.g. in sahib,

hakeem, h (S) is final consonant
li, like final h (A).

i, as in pin. I, as in pique.

k, as in kick.

kh, (A) the hard ch of Scotch loch,
German acli, especially of German
as pronounced by the Swiss.

m, n, (S) like French final m or n,
nasalizing the preceding vowel.

n, see d. n, like ng in singing.

o, (S) as in so. (A) as in obey.

q, (A) like k (or c) in cook ; further
back in the mouth than in kick. •

r, (S) English r, smooth and un-
trilled. (A) stronger. r,(S)rused
as vowel, as in apron when pro-
nounced aprn and not apern\ mod-
ern Hindus say ri, hence our am-
rita, Krishna, for a-mrta, Krsna.

s, as in same, s, see d. S, (S) Eng-
lish sh (German sell) .

t, see d.

u, as in put. u, as in rule.

y, as in you.

z, see d.

(A) a sound kindred to the spiritus
lenis (that is, to our ears, the mere
distinct separation of a vowel from
the preceding sound, as at the be-
ginning of a word in German) and
to h. The * is a very distinct sound
in Arabic, but is more nearly
represented by the spiritus lenis
than by any sound that we can
produce without much special
training. That is, it should be
treated! as silent, but the sounds
that precede and follow it should
not run together. In Arabic words
adopted into English it is treated
as silent, e.g. in Arab, amber,
Caaba ('Arab, ‘anbar, ka'abah).

(A) A final long vowel is shortened before al (7) or ibn (whose i is then
silent) .

Accent : (S) as if Latin ; in determining the place of the accent m and
u count as consonants, but h after another consonant does not. (A), on
the last syllable that contains a long vowel or a vowel followed by two
consonants, except that a final long vowel is not ordinarily accented; if
there is no long vowel nor two consecutive consonants, the accent falls on
the first syllable. The words al and ibn are never accented.

THE

HINDU -ARABIC NUMERALS

CHAPTER I

EARLY IDEAS OF THEIR ORIGIN

It lias long been recognized that the common numerals
used in daily life are of comparatively recent origin.
The number of systems of notation employed before
the Christian era was about the same as the number of
written languages, and in 'some cases a single language
had several systems. The Egyptians, for example, had
three systems of writing, with a numerical notation for
each ; the Greeks had two well-defined sets of numerals,
and the Roman symbols for number changed more or less
from century to century. Even to-day the number of
methods of expressing numerical concepts is much
greater than one would believe before making a study
of the subject, for the idea that our common numerals
are universal is far from being correct. It will be well,
then, to think of the numerals that we still commonly
call Arabic, as only one of many systems in use just
before the Christian era. As it then existed the system
was no better than many others, it was of late origin, it
contained no zero, it was cumbersome and little used,

THE IIINDU-ARABIC NUMERALS

and it had no particular promise. N ot until centuries later
did the system have any standing in the world of busi-
ness and science ; and had the place value which now
characterizes it, and which requires a zero, been worked
out in Greece, we might have been using Greek numerals
to-day instead of the ones with which we are familiar.

Of the first number forms that the world used this is
not the place to speak. Many of them are interesting,
but none had much scientific value. In Europe the in-
vention of notation was generally assigned to the eastern
shores of the Mediterranean until the critical period of
about a century ago, — sometimes to the Hebrews, some-
times to the Egyptians, but more often to the early
trading Phoenicians.1

The idea that our common numerals are Arabic in
origin is not an old one. The mediaeval and Renaissance
writers generally recognized them as Indian, and many
of them expressly stated that they were of Hindu origin.2

1 “ Discipulus. Quis primus invenit numerum apud Ilebrseos et
JEgyptios ? Magister. Abraham primus invenit numerum apud
Hebneos, deinde Moses ; et Abraham tradidit istam scientiam numeri
ad vEgyptios, et docuit eos : deinde Josephus.” [Bede, De computo
dialogus (doubtfully assigned to him), Opera omnia , Paris, 1862, Vol. I,
p. 650.]

“ Alii referunt ad Phoenices inventores arithmetical, propter eandem
commerciorum caussam : Alii ad Indos : Ioannes de Sacrobosco, cujus
sepulchrum est Lutetise in comitio Maturinensi, refert ad Arabes.”
[Ramus, Arithmetical libri dvo , Basel, 1569, p. 112.]

Similar notes are given by Peletarius in his commentary on the
arithmetic of Gemma Frisius (1563 ed., fol. 77), and in his own work
(1570 Lyons ed., p. 14) : “La valeur des Figures commence au coste
dextre tirant vers le coste senestre : au rebours de notre maniere
d’escrire par ce que la premiere prattique est venue des Chaldees :
ou des Pheniciens, qui out 6t6 les premiers traffiquers de marohan-
dise.”

2 Maximus Planudes (c. 1330) states that “ the nine symbols come
from the Indians.” [Wasclike’s German translation, Halle, 1878,

EARLY IDEAS OF TIIEIR ORIGIN

Others argued that they were probably invented by the
Chaldeans or the Jews because they increased in value
from right to left, an argument that would apply quite
as well to the Roman and Greek systems, or to any
other. It was, indeed, to the general idea of notation
that many of these writers referred, as is evident from
the words of England’s earliest arithmetical textbook-
maker, Robert Recorde (c. 1542): “In that thinge all
men do agree, that the Chaldays, whiche fyrste inuented
thys arte, did set these figures as thei set all their letters,
for they wryte backwarde as you tearme it, and so doo
they reade. And that may appeare in all Hebrewe,
Clialdaye and Arabike bookes . . . where as the Greekes,
Latines, and all nations of Europe, do wryte and reade
from the lefte hand towarde the ryghte.” 1 Others, and

p. 3.] Willichius speaks of the “ Zyphrje Indicae,” in his Arithmetical
libri tres (Strasburg, 1540, p. 93), and Cataneo of “le noue figure de
gli Indi,” in his Le pratiche delle dve prime mathematiche (Venice, 1540,
fol. 1). Woepcke is not correct, therefore, in saying (“M&noire sur la
propagation des chiffres indiens,” hereafter referred to as Propagation
[Journal Asiatique , Vol. I (6), 1863, p. 34]) that Wallis (A Treatise on
Algebra , both historical and practical , London, 1685, p. 13, and De
algebra tractatus, Latin edition in his Opera omnia, 1693, Vol. II,
p. 10) was one of the first to give the Hindu origin.

1 From the 1558 edition of The Grovnd of Artes, fol. C, 5. Similarly
Bishop Tonstall writes : “ Qui a Chaldeis primum in finitimos, deinde
in omnes pene gentes fluxit. . . . Numerandi artem a Chaldeis esse
profectam : qui dum scribunt, a dextra incipiunt, et in leuam pro-
grediuntur.” [De arte supputandi, London, 1522, fol. B, 3.] Gemma
Frisius, the great continental rival of Recorde, had the same idea :
“Primum autem appellamus dexterum locum, eo qu6d liaec ars vel k
Chaldseis, vel ab Hebrseis ortum habere credatur, qui etiam eo ordine
scribunt” ; but this refers more evidently to the Arabic numerals.
[Arithmetical practical methodvs facilis, Antwerp, 1540, fol. 4 of the
1563 ed.] Sacrobosco (c. 1225) mentions the same thing. Even the
modern Jewish writers claim that one of their scholars, Mashallah
(c. 800), introduced them to the Mohammedan world. [C. Levias,
The Jewish Encyclopedia , New York, 1905, Vol. IX, p. 348.]

THE HIND U-AR ABIC NUMERALS

among them such influential writers as Tartaglia1 in
Italy and Kobel 2 hi Germany, asserted the Arabic origin
of the numerals, while still others left the matter unde-
cided 3 or simply dismissed them as “ barbaric.” 4 Of
course the Arabs themselves never laid claim to the in-
vention, always recognizing them indebtedness to the
Hindus both for the numeral forms and for the distin-
guishing feature of place value. Foremost among these
writers was the great master of the golden age of Bag-
dad, one of the first of the Arab writers to collect the
mathematical classics of both the East and the W est, pre-
serving them and finally passing them on to awakening
Europe. This man was Mohammed the Son of Moses,
from Khowarezm, or, more after the manner of the Arab,
Mohammed ibn Musa al-Khowarazmi,6 a man of great

1 & que esto fu trouato di fare dagli Arabi condiece figure.”
[La prima parte del general trattato di nvmeri, et misvre , Venice, 1556,
fol. 9 of the 1592 edition.]

2 “Vom welclien Arabischen auch disz Kunst entsprungen ist.”
[Ain nerv geordnet Rechenbiechlin, Augsburg, 1514, fol. 13 of the 1531
edition. The printer used the letters rv for w in “new” in the first
edition, as he had no w of the proper font.]

8 Among them Glareanus : “ Characteres simplices sunt nouem sig-
nificatiui, ab Indis usque, siue Cliahkeis asciti .1.2. 3. 4.5.6. 7. 8. 9. Est
item imus .0 circulus, qui nihil significat.” [Be YI. Arithmeticae
practicae speciebvs , Paris, 1539, fol. 9 of the 1543 edition.]

4 “ Barbarische Oder gemeine Ziffern.” [Anonymous, Das EinmaJd
Eins cum notis variorum , Dresden, 1703, p. 3.] So Vossius ( De universae
matheseos natura et constitutione liber, Amsterdam, 1650, p. 34) calls
them “ Barbaras numeri notas.” The word at that time was possibly
synonymous with Arabic.

6 His full name was 'Abu 'Abdallah Mohammed ibn Musa al-
Khowarazml. He was born in Khowarezm, “the lowlands,” the
country about the present Khiva and bordering on the Oxus, and
lived at Bagdad under the caliph al-Mamun. He died probably be-
tween 220 and 230 of the Mohammedan era, that is, between 835 and
845 a.d., although some put the date as early as 812. The best ac-
count of this great scholar may be found in an article by C. Nallino,
“Al-Huwarizml,” in the Atti della R.Accad. dei Lincei, Rome, 1896. See

EARLY IDEAS OF THEIR ORIGIN

learning and one to whom the world is much indebted
for its present knowledge of algebra 1 and of arithmetic.
Of him there will often be occasion to speak ; and in the
arithmetic which he wrote, and of which Adelhard of
Bath2 (c. 1130) may have made the translation or para-
phrase,3 he stated distinctly that the numerals were due
to the Hindus.4 This is as plainly asserted by later Arab

also Verhandlungen des 5. Congresses der Orientalisten , Berlin, 1882,
Vol. II, p. 19 ; W. Spitta-Bey in the Zeitschrift der deutschen Morgen-
land. Gesellschaft , Yol. XXXIII, p. 224; Steinschneider in the Zeit-
schrift der deutschen Morgenland. Gesellschaft, Yol. L, p. 214; Treutlein
in the Abhandlungen zur Geschichte der Mathematik, Vol. I, p. 5 ; Suter,
“ Die Mathematiker unci Astronomen der Araber und ihre Werke,”
Abhandlungen zur Geschichte der Mathematik,Yo\. X, Leipzig, 1900, p. 10,
and “Nachtrage,” in Vol. XIV, p. 158 ; Cantor, Geschichte der Mathe-
matik, Vol. I, 3ded.,pp. 712-733 etc.; F.Woepcke in Propagation, p.489.
So recently has he become known that Heilbronner, writing in 1742,
merely mentions him as “ Ben-Musa, inter Arabes Celebris Geometra,
scripsit de figuris planis & sphericis.” [ Uistoria matheseos universce,
Leipzig, 1742, p. 438.]

In this work most of the Arabic names will be transliterated sub-
stantially as laid down by Suter in his work Die Mathematiker etc.,
except where this violates English pronunciation. The scheme of pro-
nunciation of oriental names is set forth in the preface.

1 Our word algebra is from the title of one of his works, Al-jabr wa'l-
muqabalah, Completion and Comparison. The work was translated into
English by F. Rosen, London, 1831, and treated in L'Algblyre d'al-
Khdrizmi et les mAthodes indienne et grecque , L£on Rodet, Paris, 1878,
extract from the Journal Asiatique. For the derivation of the word
algebra, see Cossali, Scritti Inediti, pp. 381-383, Rome, 1857 ; Leo-
nardo’s Liber Abbaci (1202), p. 410, Rome, 1857 ; both published by B.
Boncompagni. “Almuchabala” also was used as a name for algebra.

2 This learned scholar, teacher of O’Creat who wrote the Uelceph
( uPrologus N. Ocreati in Uelceph ad Adelardum Batensem magistrum
mum"), studied in Toledo, learned Arabic, traveled as far east as
Egypt, and brought from the Levant numerous manuscripts for study
and translation. See Henry in the Abhandlungen zur Geschichte der
Mathematik, Vol. Ill, p. 131 ; Woepcke in Propagation, p. 518.

3 The title is Algoritmi de numero Indorum. That he did not make
this translation is asserted by Enestroin in the Bibliotheca Mathematica,
Vol. I (3), p. 520.

4 Thus he speaks “de numero indorum per .IX. literas,” and pro-
ceeds : “ Dixit algoritmi : Cum uidissem yndos constituisse .IX. literas

. THE IIINDU-ARABIC NUMERALS

writers, even to the present clay.1 Indeed the plirase
'ilm liindi, “Indian science,” is used by them for arith-
metic, as also the adjective hindl alone.2

Probably the most striking testimony from Arabic
sources is that given by the Arabic traveler and scholar
Mohammed ibn Ahmed, Abu ’1-Rlhan al-B Irani (973-
1048), who spent many years in Hindustan. He wrote
a large work on India,3 one on ancient chronology,4 the
“ Book of the Ciphers,” unfortunately lost, which treated
doubtless of the Hindu art of calculating, and was the
author of numerous other works. Al-Blrunl was a man
of unusual attainments, being versed in Arabic, Persian,
Sanskrit, Hebrew, and Syriac, as well as in astronomy,
chronology, and mathematics. In his work on India he
gives detailed information concerning the language and

in uniuerso numero suo, propter dispositionem suam quain posuerunt,
uolui patefacere de opera quod fit per eas aliquid quod esset leuius
discentibus, si deus uoluerit.” [Boncompagni, Trattati d' Aritmetica,
Rome, 1857.] Discussed by F. Woepcke, Sur V introduction de Varith-
mAtique indienne en Occident , Rome, 1859.

1 Thus in a commentary by 'All ibn Abi Bekr ibn al-Jamal al- Ansar!
al-Mekkl on a treatise on gobar arithmetic (explained later) called Al-
mursliidah, found by Woepcke in Paris ( Propagation , p. 6G), there is
mentioned the fact that there are “nine Indian figures” and “a sec-
ond kind of Indian figures . . . although these are the figures of the
gobar writing.” So in a commentary by Hosein ibn Mohammed al-
Mahalll (died in 1756) on the Mokhtasar fi’ilm el-hisah (Extract from
Arithmetic) by 'Abdalqadir ibn 'All al-Sakhawi (died c. 1000) it is re-
lated that “ the preface treats of the forms of the figures of Hindu
signs, such as were established by the Hindu nation.” [Woepcke,
Propagation , p. 63.]

2 See also Woepcke, Propagation , p. 505. The origin is discussed at
much length by G. R. Kaye, “Notes on Indian Mathematics. Arith-
metical Notation,” Journ. and Proc. of the Asiatic Soc. of Bengal, Vol.
Ill, 1907, p. 489.

3 Alberuni's India, Arabic version, London, 1887 ; English transla-
tion, ibid., 1888.

4 Chronology of Ancient Nations, London, 1879. Arabic and English
versions, by C. E. Sachau.

EARLY IDEAS OF TIIEIll ORIGIN .

customs of the people of that country, and states ex-
plicitly 1 that the Hindus of his time did not use the
letters of then* alphabet for numerical notation, as the
Arabs did. He also states that the numeral signs called
ahka 2 had different shapes in various parts of India, as
was the case with the letters. In his Chronology of An-
cient Nations he gives the sum of a geometric progression
and shows how, in order to avoid any possibility of error,
the number may be expressed in three different systems :
with Indian symbols, in sexagesimal notation, and by an
alphabet system which will be touched upon later. He
also speaks3 of “179, 876, 755, expressed in Indian
ciphers,” thus again attributing these forms to Hindu
sources.

Preceding Al-Biruni there was another Arabic writer
of the tenth century, Motahhar ibn Tahir,4 author of
the Book of the Creation and of History , who gave as a
curiosity, in Indian (Nagari) symbols, a large number
asserted by the people of India to represent the duration
of the world. Huart feels positive that in Motahhar’s
time the present Arabic symbols had not yet come into
use, and that the Indian symbols, although known to
scholars, were not current. Unless this were the case,
neither the author nor his readers would have found
anything extraordinary in the appearance of the number
which he cites.

Mention should also be made of a widely-traveled
student, Al-Mas'udi (885 ?-956), whose journeys carried
him from Bagdad to Persia, India, Ceylon, and even

1 India , Vol. I, chap. xvi.

2 The Hindu name for the symbols of the decimal place system.

3 Sachau’s English edition of the Chronology, p. (54.

4 Literature arabe, Cl. Huart, Paris, 1902.

THE HINDU-ARABIC NUMERALS

across the China sea, and at other times to Madagascar,
Syria, and Palestine.1 He seems to have neglected no
accessible sources of information, examining also the
history of the Persians, the Hindus, and the Romans.
Touching the period of the Caliphs his work entitled
Meadows of Grold furnishes a most entertaming fund of
information. He states2 that the wise men of India,
assembled by the king, composed the Sindhind. Fur-
ther on3 he states, upon the authority of the historian
Mohammed ibn 'All 'Abdl, that by order of Al-Mansur
many works of science and astrology were translated into
Arabic, notably the Sindhind ( Siddhanta ). Concerning
the meaning and spelling of this name there is consider-
able diversity of opinion. Colebrooke 4 first pointed out
the connection between Siddhanta and Sindhind. He
ascribes to the word the meaning “ the revolving ages.” 5
Similar designations are collected by Sedillot,6 who in-
clined to the Greek origin of the sciences commonly
attributed to the Hindus.7 Casiri,8 citing the Tdrxkh al-
hokama or Chronicles of the Learned ,9 refers to the work

1 Huart, History of Arabic Literature , English ed., New York, 1903,
p. 182 seq.

2 Al-Mas'udi’s Meadows of Gold, translated in part by Aloys Spren-
ger, London, 1841 ; Les prairies dor, trad, par C. Barbier de Meynard
et Pavet de Courteille, Yols. I to IX, Paris, 1801-1877.

3 Les prairies d’or, Yol. VIII, p. 289 seq.

4 Essays , Vol. II, p. 428.

6 Loc. cit., p. 504.

G MaUriaux pour servir a Vhistoire comparcc des sciences mathema-
tiques chez les Grecs et les Orientaux, 2 vols., Paris, 1845-1849, pp. 438-
439.

7 He made an exception, however, in favor of the numerals, loc. cit.,
Vol. II, p. 503.

8 Bibliotheca Arabico-Hispana Escurialensis, Madrid, 1700-1770,
pp. 420-427.

9 The author, Ibn al-Qifti, flourished a.p. 1198 [Colebrooke, loc. cit.,
note Vol. II, p 510].

EARLY IDEAS OF THEIR ORIGIN

as the Sindum-Indum with the meaning “perpetuum
aeternumque.’' The reference 1 in this ancient Arabic
work to Al-KhowarazmI is worthy of note.

This Sindhind is the book, says Mas'udI,2 which gives
all that the Hindus know of the spheres, the stars, arith-
metic,3 and the other branches of science. He mentions
also Al-Khowarazmi and Id abash 4 * as translators of the
tables of the Sindhind. Al-Birunl6 refers to two other
translations from a work furnished by a Hindu who
came to Bagdad as a member of the political mission
which Sindh sent to the caliph Al-Mansur, in the year of
the Hejira 154 (a.d. 771).

The oldest work, in any sense complete, on the history
of Arabic literature and history is the Kitab al-Fihrist ,
written in the year 987 a.d., by Ibn Abi Ya'qub al-Nadim.
It is of fundamental importance for the history of Arabic
culture. Of the ten chief divisions of the work, the sev-
enth demands attention in this discussion for the reason
that its second subdivision treats of mathematicians and
astronomers.6

1 “Liber Artis Logisticae k Mohamaclo Ben Musa AUchuarezmila
exornatus, qui ceteros omnes brevitate methodi ac facilitate praestat,
Indorum que in praeclarissimis inventis ingenium & acumen osten-
dit.” [Casiri, loc. cit., p. 427.]

2 Ma<joudi, Le lime de l' avertissement et de la revision. Translation
by B. Carra de Vaux, Paris, 1896.

8 Verifying the hypothesis of Woepcke, Propagation, that the Sind-
hind included a treatment of arithmetic.

4 Ahmed ibn 'Abdallah, Suter, Pie MathemaliJcer, etc., p. 12.

6 India, Vol. II, p. 15.

6 See H. Suter, “Das Mathematiker-Verzeichniss im Fihrist,”
Abhandlungen zur Geschichte der Mathematik, Vol. VI, Leipzig, 1892.
For further inferences to early Arabic winters the reader is referred
to H. Suter, Die Mathematiker und Astronomen der Araber und Hire
Werke. Also “Nachtrage und Berichtigungen” to the same ( Abhand-
lungen, Vol. XIV, 1902, pp. 155-180).

THE HINDU-ARABTC NUMERALS

The first of the Arabic writers mentioned is Al-KindT
(800-870 A. d.), who wrote five books on arithmetic and
four books on the use of the Indian method of reckoning.
Sened ibn 'All, the Jew, who was converted to Islam under
the caliph Al-Mamun, is also given as the author of a work
on the Hindu method of reckoning. Nevertheless, there
is a possibility 1 that some of the works ascribed to Sened
ibn 'All are really works of Al-Ivhowarazml, whose name
immediately precedes his. However, it is to be noted in
this connection that Casiri 2 also mentions the same writer
as the author of a most celebrated work on arithmetic.

To Al-SufI, who died in 986 a.d., is also credited a large
work on the same subject, and similar treatises by other
writers are mentioned. We are therefore forced to the
conclusion that the Arabs from the early ninth century
on fully recognized the Hindu origin of the new numerals.

Leonard of Pisa, of whom we shall speak at length in
the chapter on the Introduction of the Numerals into
Europe, wrote his Liber Abbaei 3 in 1202. In this work
he refers frequently to the nine Indian figures,4 thus
showing again the general consensus of opinion in the
Middle Ages that the numerals were of Hindu origin.

Some interest also attaches to the oldest documents on
arithmetic in our own language. One of the earliest

1 Suter, loc. cit., note 165, pp. 62-63.

2 “ Send Ben Ali, . . . turn arithmetica scripta maxima celebrata,
quae publici juris fecit.” [Loc. cit., p. 440.]

8 Scritti di Leonardo Pisano , Vol. I, Liber Abbaei (1857); Yol. II,
Scritti (1862); published by Baldassarre Boncompagni, Rome. Also
Tre Scritti Inediti, and Intorno ad Opere di Leonardo Pisano , Rome,
1854.

4 “ Ubi ex mirabili magisterio in arte per novem flguras indorum
introductus” etc. In another place, as a heading to a separate divi-
sion, he writes, “De cognitione novem figurarum yndorum” etc.
“ Novem figure indorum lie sunt 987 654321.”

EARLY IDEAS OF TIIETR ORIGIN

treatises on algorism is a commentary 1 on a set of
verses called the Carmen de Algorismo, written by Alex-
ander de Villa Dei (Alexandre de Ville-Dieu), a Minor-
ite monk of about 1240 a.d. The text of the first few
lines is as follows :

“Idee algorism’ ars p’sens dicit’ in qua
Talib; indor^ fruim bis quinq; figuris.2

“This boke is called the boke of algorim or augrym
after lewder use. And this boke tretys of the Craft of
Nombryng, the quych crafte is called also Algorym.
Tlier was a kyng of Inde the quich heyth Algor & he
made tills craft. . . . Algorisms, in the quych we use
teen figurys of Inde.”

1 See An Ancient English Algorism , by Davicl Eugene Smith, in
Festschrift Moritz Cantor , Leipzig, 1909. See also Victor Mortet, “Le
plus ancien traits francais d’algorisme,” Bibliotheca Mathematica, Vol.
IX (3), pp. 55-64.

2 These are the two opening lines of the Carmen de Algorismo that
the anonymous author is explaining. They should read as follows :

Haec algorismus ars praesens dicitur, in qua
Talibus Indorum fruimur his quinque figuris.

What follows is the translation.

CHAPTER II

EARLY HINDU FORMS WITH NO PLACE VALUE

While it is generally conceded that the scientific de-
velopment of astronomy among the Hindus towards
the beginning of the Christian era rested upon Greek 1
or Chinese 2 sources, yet their ancient literature testifies
to a high state of civilization, and to a considerable ad-
vance in sciences, in philosophy, and along literary lines,
long before the golden age of Greece. From the earliest
tunes even up to the present day the Hindu has been
wont -to put his thought into rhythmic form. The first
of this poetry — it well deserves this name, being also
worthy from a metaphysical point of view 3 — consists of
the Vedas, hymns of praise and poems of worship, col-
lected during the V edic period which dates from approxi-
mately 2000 b.c. to 1400 b.c.4 Following this work, or
possibly contemporary with it, is the Brahmanic literature,
which is partly ritualistic (the Brahmanas), and partly
philosophical (the Upanisliads). Our especial interest is

1 Thibaut, Astronomic, Astrologie und MathematiJc, Strassburg,
1899.

2 Gustave Schlegel, Uranographie cliinoise ou preuves directes que
V astronomic primitive est originaire de la Chine , et qu'elle a 6tA emprun-
tAe par les anciens peuples occidentaux d la sphere cliinoise ; ouvrage ac-
compagnd d'un atlas cdleste chinois et grec , The Hague and Leyden,
1875.

8 E. W. Hopkins, The Religions of India , Boston, 1898, p. 7.

4 It. C. Dutt, History of India, London, 1906.

EARLY HINDU FORMS WITH NO PLACE VALUE 13

in the Sutras, versified abridgments of the ritual and of
ceremonial rules, which contain considerable geometric
material used hi connection with altar construction, and
also numerous examples of rational numbers the sum of
whose squares is also a square, i.e. “ Pythagorean num-
bers,” although this was long before Pythagoras lived.
Whitney 1 places the whole of the V eda literature, includ-
ing the Vedas, the Brahmanas, and the Sutras, between
1500 b.c. and 800 b.c., thus agreeing with Burk 2 who
holds that the knowledge of the Pythagorean theorem re-
vealed in the Sutras goes back to the eighth century b.c.

The importance of the Sutras as showing an independ-
ent origin of Hindu geometry, contrary to the opinion
long held by Cantor 3 of a Greek origin, has been repeat-
edly emphasized in recent literature,4 especially since
the appearance of the important work of Von Schroeder.6
Further fundamental mathematical notions such as the
conception of irrationals and the use of gnomons, as well as
the philosophical doctrine of the transmigration of souls,

— all of these having long been attributed to the Greeks,

— are shown in these works to be native to India. Al-
though this discussion does not bear directly upon the

1 W. D. Whitney, Sanskrit Grammar , 3d ed., Leipzig, 1896.

2 “ Das Apastainba-Sulba-Sutra,” Zeitschrift der deutschen Morgen-
landischen Gesellschaft, Vol. LV, p. 543, and Vol. LVI, p. 327.

3 Geschichte der Math., Vol. I, 2d ed., p. 595.

4 L. von Schroeder, Pythagoras und die Index , Leipzig, 1884 ; H.
Vogt, “ Haben die alten Inder den Pythagoreischen Lehrsatz und das
Irrationale gekannt? ” Bibliotheca Mathematica, Vol. VII (3), pp. 6-20;
A. Burk, loc. cit. ; Max Simon, Geschichte der Mathematik im Altertum,
Berlin, 1909, pp. 137-165 ; three Sutras are translated in part by
Thibaut, Journal of the Asiatic Society of Bengal , 1875, and one ap-
peared in The Pandit, 1875; Beppo Levi, “ Osservazioni e congetturc
sopralageometriadegli indiani,” Bibliotheca Mathematica, Vol. IX (3),
1908, pp. 97-105.

6 Loc. cit.; also Indiens Literatur und Cultur, Leipzig, 1887,

THE IIINDU-ARABIC NUMERALS

origin of our numerals, yet it is highly pertinent as show-
ing the aptitude of the Hindu for mathematical and men-
tal work, a fact further attested by the independent
development of the drama and of epic and lyric poetry.

It should be stated definitely at the outset, however,
that we are not at all sure that the most ancient forms
of the numerals commonly known as Arabic had their
origin in India. As will presently be seen, their forms
may have been suggested by those used in Egypt, or in
Eastern Persia, or in China, or on the plains of Mesopo-
tamia. W e are quite in the dark as to these early steps ;
but as to their development in India, the approximate
period of the rise of their essential feature of place value,
their introduction into the Arab civilization, and then-
spread to the West, we have more or less definite infor-
mation. When, therefore, we consider the rise of the
numerals in the land of the Sindliu,1 it must be under-
stood that it is only the large movement that is meant,
and that there must further be considered the numerous
possible sources outside of India itself and long anterior
to the first prominent appearance of the number symbols.

No one attempts to examine any detail in the lnstory of
ancient India without being struck with the great dearth
of reliable material.2 So little sympathy have the people
with any save those of their own caste that a general litera-
ture is wholly lacking, and it is only in the observations
of 'strangers that any all-round view of scientific progress
is to be found. There is evidence that primary schools

1 It is generally agreed that the name of the river Sindliu, corrupted
by western peoples to Hindliu, Indos, Indus, is the root of Hindustan
and of India. Reclus, Asia , English ed., Vol. Ill, p. 14.

2 See the comments of Oppert, On the Original Inhabitants of Bham-
lavarsa or India, London, 1893, p, 1,

EARLY HINDU FORMS WITH NO PLACE VALUE 15

existed in earliest times, and of the seventy-two recognized
sciences writing and arithmetic were the most prized.1 In
the Vedic period, say from 2000 to 1400 B.c., there was
the same attention to astronomy that was found in the
earlier civilizations of Babylon, China, and Egypt, a fact at-
tested by the Vedas themselves.2 Such advance in science
presupposes a fair knowledge of calculation, but of the
manner of calculating we are quite ignorant and prob-
ably always shall be. One of the Buddhist sacred books,
the Lalitavistara, relates that when the Bodhisattva 3 was
of age to marry, the father of Gopa, his intended bride,
demanded an examination of the five hundred suitors,
the subjects including arithmetic, writing, the lute, and
archery. Having vanquished his rivals hi all else, he is
matched against Arjuna the great arithmetician and is
asked to express numbers greater than 100 kotis.4 In
reply he gave a scheme of number names as high as 1063,
adding that he could proceed as far as 10421,6 all of which
suggests the system of Archimedes and the unsettled
question of the indebtedness of the West to the East in
the realm of ancient mathematics.6 Sir Edwin Arnold,

1 A. Hillebrandt, Alt-Indien, Breslau, 1899, p. 111. Fragmentary
records relate that Kharavela, king of Kalinga, learned as a boy lekhd
(writing), ganand (reckoning), and rupa (arithmetic applied to mone-
tary affairs and mensuration), probably in the 5th century b.c.
[Biihler, Indische Palaeographie , Strassburg, 1896, p. 5.]

2 R. C. Dutt, A History of Civilization in Ancient India, London,
1893, Vol.I, p. 174.

8 The Buddha. The date of his birth is uncertain. Sir Edwin Ar-
nold put it c. 620 b.c.

* I.e. 100-107.

6 There is some uncertainty about this limit.

0 This problem deserves more study than has yet been given it. A
beginning may be made with Comte Goblet d’Alviella, Ce que Vlnde
doit ii la Grbce , Paris, 1897, and II. G. Keene’s review, “ The Greeks in
India,” jn the Calcutta Review, Vol. CXIV, 1902, p. 1. See also F.

THE IIINDU-ARABIC NUMERALS

in The Light of Asia , does not mention this part of the
contest, but he speaks of Buddha’s training at the hands
of the learned Visvamitra :

“ And Viswamitra said, ‘ It is enough,

Let us to numbers. After me repeat
Your numeration till we reach the lakh,1
One, two, three, four, to ten, and then by tens
To hundreds, thousands.’ After him the child
Named digits, decads, centuries, nor paused,

The round lakh reached, but softly murmured on,

Then comes the koti, nahut, niunahut,

Khamba, viskliamba, abab, attata,

To kumuds, gundhikas, and utpalas,

By pundarikas into padumas,

Which last is how you count the utmost grains
Of Hastagiri ground to finest dust ; 2
But beyond that a numeration is,

The Katha, used to count the stars of night,

The Koti-Katha, for the ocean drops ;

Ingga, the calculus of circulars ;

Sarvanikchepa, by the which you deal

With all the sands of Gunga, till we come

To Antah-Kalpas, where the unit is

The sands of the ten crore Gungas. If one seeks

More comprehensive scale, th’ aritlimic mounts

By the Asankya, which is the tale

Of all the drops that in ten thousand years

Would fall on all the worlds by daily rain;

Thence unto Maha Kalpas, by the which
The gods compute their future and their past.’ ”

Woepcke, Propagation , p. 253; G. R. Kaye, loc. cit., p. 475 seq., and
“The Source of Hindu Mathematics,” Journal of the Royal Asiatic
Society, July, 1910, pp. 749-700; G. Thibaut, Astronomie, Astrologie
und Mathematik, pp. 43-50 and 70-79. It will be discussed more fully
in Chapter YI.

1 I.e. to 100,000. The lakh is still the common large unit in India,
like the myriad in ancient Greece and the million in the West.

2 This again suggests the Psammites, or De harenae numero as it is
called in the 1544 edition of the Opera of Archimedes, a work in which
the great Syracusan proposes to show to the king “ by geometric proofs
which you can follow, that the numbers which have been named by

EARLY HINDU FORMS WITH NO PLACE VALUE 17

Thereupon Visvamitra Aearya 1 expresses his approval
of the task, and asks to hear the “ measure of the line ”
as far as yojana, the longest measure bearing name. This
given, Buddha adds :

. . . “ ‘ And master ! if it please,

I shall recite how many sun-motes lie
From end to end within a yojana.’

Thereat, with instant skill, the little prince
Pronounced the total of the atoms true.

But Viswamitra heard it on his face
Prostrate before the boy ; ‘ For thou,’ he cried,

‘ Art Teacher of thy teachers — thou, not I,

Art Guru.’ ”

It is needless to say that this is far from being history.
And yet it puts in charming rhythm only what the ancient
Lalitavistara relates of the number-series of the Buddha’s
time. While it extends beyond all reason, nevertheless
it reveals a condition that would have been impossible
unless arithmetic had attained a considerable degree of
advancement.

To this pre-Christian period belong also the Vedahgas ,
or “limbs for supporting the Veda,” part of that great
branch of Hindu literature known as Smriti (recollec-
tion), that which was to be handed down by tradition.
Of these the sixth is known as Jyotisa (astronomy), a
short treatise of only thirty-six verses, written not earlier
than 300 B.C., and affording us some knowledge of the
extent of number work in that period.2 The Hindus

us . . . are sufficient to exceed not only the number of a sand-heap as
large as the whole earth, but one as large as the universe.” For a
list of early editions of this work see D. E. Smith, Bara Arithmetical
Boston, 1909, p. 227. 1 I.e. the Wise.

2 Sir Monier Monier-Williams, Indian Wisdom , 4th ed., London,
1893, pp. 144, 177. See also J. C. Marshman, Abridgment of the History
of India , London, 1893, p. 2.

THE HINDU-ARABIC NUMERALS

also speak of eighteen ancient Siddhantas or astronomical
works, which, though mostly lost, confirm this evidence.1

As to authentic histories, however, there exist in India
none relating to the period before the Mohammedan era
(622 A.D.). About all that we know of the earlier civi-
lization is what we glean from the two great epics, the
Mahabharata 2 and the Ram ay an a, from coins, and from
a few inscriptions.3

It is with this unsatisfactory material, then, that we
have to deal in searching for the early history of the
Hindu-Arabic numerals, and the fact that many unsolved
problems exist and will continue to exist is no longer
strange when we consider the conditions. It is rather
surprising that so much has been discovered within a
century, than that we are so uncertain as to origins and
dates and the early spread of the system. The probabil-
ity being that writing was not introduced into India
before the close of the fourth century b.c., and literature
existing only in spoken form prior to that period,4 the
number work was doubtless that of all primitive peoples,
palpable, merely a matter of placing sticks or cowries or
pebbles on the ground, of marking a sand-covered board,
or of cutting notches or tying cords as is still done hi
parts of Southern India to-day.5

1 For a list and for some description of these works see R. C. Dutt,
A History of Civilization in Ancient India , Vol. II, p. 121.

2 Professor Ramkrishna Gopal Bhandarkar fixes the date as the
fifth century b.c. [“Consideration of the Date of the Mahabharata,”
in the Journal of the Bombay Brandi of the E. A. Soc., Bombay, 1873,
Vol. X, p. 2.]

3 Marshman, loc. cit., p. 2.

4 A. C. Burnell, South Indian Palaeography, 2d ed., London, 1878,
p. 1, seq.

5 This extensive subject of palpable arithmetic, essentially the
history of the abacus, deserves to be treated in a work by itself.

EARLY HINDU FORMS WITH NO PLACE VALUE 19

The early Hindu numerals 1 may be classified into
three great groups, (1) the Kharosthi, (2) the Braliml,
and (3) the word and letter forms; and these will be
considered in order.

The Kharosthi numerals are found in inscriptions for-

• • <

merly known as Bactrian, Indo-Bactrian, and Aryan,
and appearing in ancient Gandhara, now eastern Afghan-
istan and northern Punjab. The alphabet of the language
is found in inscriptions dating from the fourth century
b.c. to the third century a.d., and from the fact that
the words are written from right to left it is assumed to
be of Semitic origin. No numerals, however, have been
found in the earliest of these inscriptions, number-names
probably having been written out in words as was the
custom with many ancient peoples. Not until the tune
of the powerful King Asoka, in the third century B.c.,
do numerals appear in any inscriptions thus far discov-
ered ; and then only in the primitive form of marks, quite
as they would be found in Egypt, Greece, Rome, or in

1 The following are the leading sources of information upon this
subject : G. Biihler, Indische Palaeographie , particularly chap, vi ;
A. C. Burnell, South Indian Palaeography, 2d ed., London, 1878, where
tables of the various Indian numerals are given in Plate XXIII ; E. C.
Bayley, “ On the Genealogy of Modern Numerals,” Journal of the Eoyal
Asiatic Society , Vol. XIV, part 3, and Vol. XV, part 1, and reprint,
London, 1882; I. Taylor, in The Academy , January 28, 1882, with a
repetition of his argument in his work The Alphabet, London, 1883,
Vol. II, p. 265, based on Bayley ; G. R. Kaye, loc. cit., in some respects
one of the most critical articles thus far published ; J. C. Fleet,
Corpus inscriptionum Indicarum, London, 1888, Vol. Ill, with fac-
similes of many Indian inscriptions, and Indian Epigraphy, Oxford,
1907, reprinted from the Imperial Gazetteer of India, Vol. II, pp. 1-88,
1907 ; G. Thibaut, loc. cit., Astronomie etc.; R. Caldwell, Comparative
Grammar of the Dravidian Languages, London, 1856, p. 262 seq.; and
Epigraphia Indica (official publication of the government of India),
Vols. I-IX. Another work of Biihler’s, On the Origin of the Indian
Brahma Alphabet, is also of value,

THE HINDU- ARABIC NUMERALS

various other parts of the world. These Asoka 1 inscrip-
tions, some thirty in all, are found in widely separated
parts of India, often on columns, and are in the various
vernaculars that were familiar to the people. Two are in
the Kharosthi characters, and the rest in some form of
BrahmI. In the Kharosthi inscriptions only four numer-
als have been found, and these are merely vertical marks
for one, two, four, and five, thus :

. I II INI Mill

In the so-called Saka inscriptions, possibly of the first
century b.c., more numerals are found, and in more
highly developed form, the right-to-left system appearing,
together with evidences of three different scales of count-
ing, — four, ten, and twenty. The numerals of this
period are as follows:

1 2 3 4 5 G 8 10

< II III X IX II X. XX ?

} 933 33} 9333 XI til

20 50 60 70 100 200

There are several noteworthy points to be observed hi
studying this system. In the first place, it is probably not
as early as that shown in the Nana Ghat forms hereafter
given, although the inscriptions themselves at Nana
Ghat are later than those of the Asoka period. The

1 The earliest work on the subject was by James Prinsep, “On the
Inscriptions of Piyadasi or Asoka, etc., Journal of the Asiatic Society
of Bengal, 1838, following a preliminary suggestion in the same journal
in 1837. See also “Asoka Notes,” by V. A. Smith, The Indian An-
tiquary, Vol. XXXVII, 1908, p. 24 seq., Vol. XXXVIII, pp. 151-159,
June, 1909 ; The Early History of India, 2d ed., Oxford, 1908, p. 154 ;
J. F. Fleet, “The Last Words of Asoka,” Journal of the Royal Asiatic
Society, October, 1909, pp. 981-1016; E. Senart, Les inscriptions de
Piyadasi , 2 vols., Paris, 1887,

EARLY HINDU FORMS WITH NO PLACE VALUE 21

four is to this system what the X was to the Roman,
probably a canceling of three marks as a workman does
to-day for five, or a laying of one stick across three others.
The ten has never been satisfactorily explained. It is
similar to the A of the KharosthI alphabet, but we have
no knowledge as to why it was chosen. The twenty is
evidently a ligature of two tens, and this in turn sug-
gested a kind of radix, so that ninety was probably writ-
ten in a way reminding one of the quatre-vingt-dix of
the French. The hundred is unexplained, although it
resembles the letter ta or tra of the Braliml alphabet with
1 before (to the right of) it. The two hundred is only
a variant of the symbol for hundred, with two vertical
marks.1

This system has many points of similarity with the
Nabatean numerals2 in use in the first centuries of the
Christian era. The cross is here used for four, and the
Kharostlu form is employed for twenty. In addition to
this there is a trace of an analogous use of a scale of
twenty. While the symbol for 100 is quite different, the
method of forming the other hundreds is the same. The
correspondence seems to be too marked to be wholly
accidental.

It is not in the KharosthI numerals, therefore, that we
can hope to find the origin of those used by us, and we
turn to the second of the Indian types, the Brahml char-
acters. The alphabet attributed to Brahma is the oldest of
the several known in India, and was used from the earliest
historic times. There are various theories of its origin,

1 For a discussion of the minor details of this system, see Biihler,
loc. cit., p. 73.

2 Julius Euting, Nabataische Inschriften aus Arabien, Berlin, 1885,
pp. 96-97, with a table of numerals.

TIIE HINDU-ARABTC NUMERALS

none of which has as yet any wide acceptance,1 although
the problem offers hope of solution in due time. The
numerals are not as old as the alphabet, or at least they
have not as yet been found in inscriptions earlier than
those in which the edicts of Asoka appear, some of these
having been incised in Brahnn as well as KharosthL As
already stated, the older writers probably wrote the num-
bers in words, as seems to have been the case in the
earliest Pali writings of Ceylon.2

The following numerals are, as far as known, the
only ones to appear in the Asoka edicts : 3

These fragments from the third century b.c., crude and
unsatisfactory as they are, are the undoubted early forms
from which our present system developed. They next
appear in the second century b.c. in some inscriptions in
the cave on the top of the Nana Ghat hill, about seventy-
five miles from Poona in central India. These inscrip-
tions may be memorials of the early Andhra dynasty of
southern India, but their chief interest lies in the numer-
als which they contain.

The cave was made as a resting-place for travelers as-
cending the hill, which lies on the road from Ivalyana to
Junar. It seems to have been cut out by a descendant

1 For the five principal theories see Buhler, loc. cit., p. 10.

2 Bayley, loc. cit., reprint p. 3.

3 Buhler, loc. cit.; Epigraphia Indica, Vol. Ill, p. 134 ; Indian An-
tiquary, Vol. VI, p. 155 seq., and Vol. X, p. 107.

2 4

50 50 200

200

EARLY HINDU FORMS WITH NO PLACE VALUE 23

of King Satavahana,1 for inside the wall opposite the en-
trance are representations of the members of his family,
much defaced, but with the names still legible. It would
seem that the excavation was made by order of a king
named Vedisiri, and “the inscription contains a list of
gifts made on the occasion of the performance of several
yagnas or religious sacrifices,” and numerals are to be
seen in no less than thirty places.2

There is considerable dispute as to what numerals are
really found in these inscriptions, owing to the difficulty
of deciphering them ; but the following, which have been
copied from a rubbing, are probably number forms : 3

1 ? « -or or

12 4 G 7 9 10 10 10

o 4 a> W

20 GO 80 100 100 100 200 400

W T T TT Ty Jcr Jo

TOO 1000 4000 G000 10,000 20,000

The inscription itself, so important as containing the
earliest considerable Hindu numeral system connected
with our own, is of sufficient interest to warrant repro-
ducing part of it in facsimile, as is done on page 24.

1 Pandit Bhagavanlal IndrajI, “ On Ancient Nagari Numeration ;
from an Inscription at Naneghat,” Journal of the Bombay Branch of the
Royal Asiatic Society, 1876, Vol. XII, p. 404.

2 lb., p. 405. He gives also a plate and an interpretation of each
numeral.

3 These may be compared with Biihler’s drawings, loc. cit. ; with
Bayley, loc. cit., p. 337 and plates ; and with Bayley’s article in the
j Encyclopaedia Britannica, 9th ed., art. “Numerals.”

TIIE HINDU- A R A B TC NUMERALS

flanaghat Inscriptions

Tlie next very noteworthy evidence of the numerals,
and this quite complete as will be seen, is found in cer-
tain other cave inscriptions dating back to the first or
second century a.d. In these, the Nasik1 cave inscrip-
tions, the forms are as follows :

- = = ^ 1 M ^ ?

1 23 4 5 G 7 8 9

cco-ce * 3 7 7 Jr

10 10 20 40 70 100 200 500

1 f T T P ft

1000 2000 3000 4000 S000 70,000

From this time on, until the decimal system finally
adopted the first nine characters and replaced the rest of
the Brahmi notation by adding the zero, the progress of
these forms is well marked. It is therefore well to present

1 E. Senart, “The Inscriptions in the Caves at Nasik,” Epigraphia
Indica, Vol. VIII, pp. 59-9G ; “The Inscriptions in the Cave at Ivarle,”
Epigraphia Indica , Vol. VII, pp. 47-74; Biihler, Palaeographie , Tafel
IX.'

Table showing the Progress of Number Forms

in India

Numerals

; 1 2

4 5 6 7 8 9 10

30 40

50 GO 70 80

90 100 200 1000

» Asoka

f II

III

III!

b Saka

Ifl

III

X IXIIX XX ?

733333

AllH

o Asoka

1 II

-h A>

d Nagarl

—

X- ©7 Po (o

^ CP

VKT

o Nasik

—

¥ P £ 7 b ? or

7 T 1

f Ksatrapa

— .

—

tt"PQZ J«-

NT X

jy x ®

s Kusana

. _

^4-F 6 o

Q> X09®

h Gupta

3cc

i ValhabI

% -

* 9*

wry 3 com T

i Nepal

A ? y

k Kaliiiga

J> V ?? d>

1 Vakataka

a KharosthI numerals, ASoka inscriptions, c. 250 n.c. Senart, Notes
d'6pigraphie indienne. Given by Btililer, loc. cit., Tafel I.

b Same, Saka inscriptions, probably of the first century b.c.
Senart, loc. cit. ; Biililer, loc. cit.

c Brahrnl numerals, A6oka inscriptions, c. 250 n.c. Indian Anti-
quary, Vol. VI, p. 155 seq.

d Same, Nana Ghat inscriptions, c. 150 b.c. Bhagavanlal Inclraji,
On Ancient Ndgari Numeration, loc. cit. Copied from a squeeze of
the original.

e Same, Nasik inscription, c. 100 b.c. Burgess, Archeological Survey
Report, Western India ; Senart, Epigraphia Indica , Vol. VII, pp. 47-
79, and Vol. VIII, pp. 59-96.

f Ksatrapa coins, c. 200 a.d. Journal of the Royal Asiatic Society,
1890, p. 639.

s Ivusana inscriptions, c. 150 a.d. Epigraphia Indica, Vol. I, p. 381,
and Vol. II, p. 201.

h Gupta Inscriptions, c. 300 a.d. to 450 a.d. Fleet, loc. cit., Vol. III.

* ValhabI, c. 600 a.d. Corpus, Vol. III.

i Bendall’s Table of Numerals, in Cat. Sansk. Budd. MSS., British
Museum.

k Indian Antiquary, Vol. XIII, 120; Epigraphia Indica, Vol. Ill,
127 ff. 1 Fleet, loc. cit.

[Most of these numerals are given by Biililer, loc. cit., Tafel IX.]

THE HINDU- ARABIC NUMERALS

synoptically the best-known specimens that have come
down to us, and this is done in the table on page 25.1

With respect to these numerals it should first be noted
that no zero appears in the table, and as a matter of fact
none existed in any of the cases cited. It was therefore
impossible to have any place value, and the numbers like
twenty, thirty, and other multiples of ten, one hundred, ,
and so on, required separate symbols except where they
were written out in words. The ancient Hindus had no
less than twenty of these symbols,2 3 4 * 6 7 8 a number that was
afterward greatly increased. The following are examples
of their method of indicating certain numbers between
one hundred and one thousand:

for 174 4<H (X) — for191

1 See Fleet, loc. cit. See also T. Benfey, Sanskrit Grammar , Lon-
don, 1803, p. 217 ; M. R. Kale, Higher Sanskrit Grammar , 2d eel., Bom-
bay, 1898, p. 110, and other authorities as cited.

2 Bayley, loc. cit., p. 335.

3 From a copper plate of 493 a.d., found at Karltalal, Central
India. [Fleet, loc. cit., Plate XVI.] It should be stated, however, that
many of these copper plates, being deeds of property, have forged
dates so as to give the appearance of antiquity of title. On the other
hand, as Colebrooke long ago pointed out, a successful forgery has
to imitate the writing of the period in question, so that it becomes
evidence well worth considering, as shown in Chapter III.

4 From a copper plate of 510 a.d., found at Majhgawain, Central

India. [Fleet, loc. cit., Plate XIV.]

6 From an inscription of 588 a.d., found at Bodh-Gaya, Bengal
Presidency. [Fleet, loc. cit., Plate XXIV.]

6 From a copper plate of 571 a.d., found at Maliya, Bombay Presi-
dency. [Fleet, loc. cit., Plate XXIV.]

7 From a Biiayagadli pillar inscription of 372 a.d. [Fleet, loc. cit.,
Plate XXXVI, C.]

8 From a copper plate of 434 a.d. [ Indian Antiquary , Vol. I, p. 60.]

EARLY HINDU FORMS WITH NO PLACE VALUE 27

To these may be added the following numerals below
one hundred, similar to those in the table :

GO 1 for 90 2 for 70

We have thus far spoken of the KharosthI and BrahmT
numerals, and it remains to mention the third type, the
word and letter forms. These are, however, so closely
connected with the perfecting of the system by the inven-
tion of the zero that they are more appropriately consid-
ered in the next chapter, particularly as they have little
relation to the problem of the origin of the forms known
as the Arabic.

Having now examined types of the early forms it is
appropriate to turn our attention to the question of their
origin. As to the first three there is no question. The
I or — is simply one stroke, or one stick laid down by
the computer. The II or = represents two strokes or
two sticks, and so for the III and E . From some primi-
tive 1 1 came the two of Egypt, of Rome, of early Greece,
and of various other civilizations. It appears in the
three Egyptian numeral systems in the following forms :

Hieroglyphic 1 1
Hieratic (I
Demotic R ^

The last of these is merely a cursive form as in the
Arabic F, which becomes our 2 if tipped through a
right angle. From some primitive = came the Chinese

1 Gadhwa inscription, c. 417 a.d. [Fleet, loc. cit., Plate IV, I).]

2 Karitalai plate of 493 a.d., referred to above.

THE IIINDU-ARABIC NUMERALS

symbol, which is practically identical with the symbols
found commonly hi India from 150 b.c. to 700 a.d. In
the cursive form it becomes z, and this was frequently
used for two in Germany until the 18th century. It
finally went into the modern form 2, and the = in the
same way became our 3.

There is, however, considerable ground for interesting
speculation with respect to these first three numerals.
The earliest Hindu forms were perpendicular. In the
Nana Ghat inscriptions they are vertical. But long before
either the Asoka or the Nana Ghat inscriptions the Chi-
nese were using the horizontal forms for the first three
numerals, but a vertical arrangement for four.1 Now
where did China get these forms ? Surely not from
India, for she had them, as her monuments and litera-
ture2 show, long before the Hindus knew them. The
tradition is that China brought her civilization around
the north of Tibet, from Mongolia, the primitive habitat
being Mesopotamia, or possibly the oases of Turkestan.
Now what numerals did Mesopotamia use? The Baby-
lonian system, simple in its general principles but very
complicated in many of its details, is now well known.3
In particular, one, two, and three were represented by
vertical arrow-heads. Why, then, did the Chinese write

1 It seems evident that the Chinese four, curiously enough called
“ eight in the mouth,” is only a cursive 1 1 1 1.

2 Chalfont, F. H., Memoirs of the Carnegie Museum, Vol. IV, no. 1 ;
J. Hager, An Explanation of the Elementary Characters of the Chinese ,
London, 1801.

3 II. V. Hilprecht, Mathematical, Metrological and Chronological
Tablets from the Temple Library at Nippur, Vol. XX, parti, of Series
A, Cuneiform Texts Published by the Babylonian Expedition of the
University of Pennsylvania, 1906; A. Eisenlohr, Ein altbabylonischer
Felderplan, Leipzig, 1906 : Maspero, Dawn of Civilization, p. 773.

EARLY HINDU FORMS WITH NO PLACE VALUE 29

theirs horizontally ? The problem now takes a new inter-
est when we find that these Babylonian forms were not
the primitive ones of this region, but that the early
Sumerian forms were horizontal.1

What interpretation shall be given to these facts ?
Shall we say that it was mere accident that one people
wrote “ one ” vertically and that another wrote it horizon-
tally ? This may be the case ; but it may also be the
case that the tribal migrations that ended in the Mongol
invasion of China started from the Euphrates while yet
the Sumerian civilization was prominent, or from some
conun on source in Turkestan, and that they carried to
the East the primitive numerals of their ancient home,
the first three, these being all that the people as a whole
knew or needed. It is equally possible that these three
horizontal forms represent primitive stick-laying, the most
natural position of a stick placed in front of a calculator
being the horizontal one. When, however, the cuneiform
writing developed more fully, the vertical form may have
been proved the easier to make, so that by the time the
migrations to the West began these were in use, and
from them came the upright forms of Egypt, Greece,
Rome, and other Mediterranean lands, and those of
Asoka’s time in India. After Asoka, and perhaps among
the merchants of earlier centuries, the horizontal forms
may have come down into India from China, thus giving
those of the Nana Ghat cave and of later inscriptions. This
is in the realm of speculation, but it is not improbable that
further epigraphical studies may confirm the hypothesis.

1 Sir H. II. Howard, “On the Earliest Inscriptions from Chaldea,”
Proceedings of the Society of Biblical Archaeology , XXI, p. 301, London,
1899.

TIIE IIINDU-ARABIC NUMERALS

As to the numerals above three there have been very
many conjectures. The figure one of the Demotic looks
like the one of the Sanskrit, the two (reversed) like that of
the Arabic, the four has some resemblance to that in the
Nasik caves, the five (reversed) to that on the Ksatrapa
coins, the nine to that of the Kusana inscriptions, and
other points of similarity have been imagined. Some
have traced resemblance between the Hieratic five and
seven and thqse of the Indian inscriptions. There have
not, therefore, been wanting those who asserted an Egyp-
tian origin for these numerals.1 There has already been
mentioned the fact that the Kharosthi numerals were
formerly known as Bactrian, Indo-Bactrian, and Aryan.
Cunningham2 was the first to suggest that these nu-
merals were derived from the alphabet of the Bactrian
civilization of Eastern Persia, perhaps a thousand years
before our era, and in this he was supported by the
scholarly work of Sir E. Clive Bayley,3 who in turn
was followed by Canon Taylor.4 The resemblance has
not proved convincing, however, and Bay ley’s drawings

1 Eor a bibliography of the principal hypotheses of this nature see
Biihler, loc. cit., p. 77. Biililer (p. 78) feels that of all these hypotheses
that which connects the Braliml with the Egyptian numerals is the
most plausible, although he does not adduce any convincing proof.
Tli. Henri Martin, “Les signes num&raux et Tarithindtique chez les
peuples de l’antiquitd et du moyen fige ” (being an examination of
Cantor’s Mathematische Beitrdge zum Culturleben derVoUcer), Annalidi
matematica pura ed applicata, Yol.Y, Rome, 18G4, pp. 8, 70. Also, same
author, “ Recherches nouvelles sur l’origine de notre syst&me de nu-
meration dcrite,” Revue ArcMologique, 1857, pp. 3G, 55. See also the
tables given later in this work.

2 Journal of the Royal Asiatic Society , Bombay Branch , Vol. XXIII.

8 Loc. cit., reprint, Part I, pp. 12, 17. Bayley’s deductions are

generally regarded as unwarranted.

4 The Alphabet , London, 1883, Vol. II, pp. 265, 2GG, and The Acad-
emy of Jan. 28, 1882.

EARLY HINDU FORMS WITH NO PLACE VALUE 31

have been criticized as being affected by his theory. The
following is part of the hypothesis : 1

Numeral Hindu

Bactrian

Sanskrit

4 ^

^ =ch

chatur, Lat. quattuor

5 h

b =P

panclia, Gk. irtvre

6 tp

y -

-j g an(j g are interchanged as

7 7

7 =8

I occasionally in N.W. India

sapta J

Btihler2 rejects this hypothesis, stating that in four
cases (four, six, seven, and ten) the facts are absolutely
against it.

While the relation to ancient Bactrian forms has been
generally doubted, it is agreed that most of the numerals
resemble Brahnff letters, and we would naturally expect
them to be initials.3 But, knowing the ancient pronunci-
ation of most of the number names,4 we find this not
to be the case. We next fall back upon the hypothesis

1 Taylor, The Alphabet, loc. cit., table on p. 266.

2 Btihler, On the Origin of the Indian Brdlima Alphabet, Strassburg,
1898, footnote, pp. 52, 53.

8 Albrecht Weber, History of Indian Literature, English eel., Bos-
ton, 1878, p. 256 : “ The Indian figures from 1-9 are abbreviated forms
of the initial letters of the numerals themselves . . . : the zero, too,
has arisen out of the first letter of the word sunya (empty) (it occurs
even in Pingala) . It is the decimal place value of these figures which
gives them significance.” C. Henry, “Sur l'origine de quelques nota-
tions math£matiques,” Revue ArcMologique , June and July, 1879, at-
tempts to derive the Boethian forms from the initials of Latin words.
See also J. Prinsep, “Examination of the Inscriptions from Girnar in
Gujerat, and Dhauli in Cuttach,” Journal of the Asiatic Society of Ben-
gal, 1838, especially Plate XX, p. 348 ; this was the first work on the'
subject.

4 Btihler, Palaeographie , p. 75, gives the list, with the list of letters
(p. 76) corresponding to the number symbols,

TIIE TIINDU-ARABIC NUMERALS

that they represent the order of letters 1 in the ancient
alphabet. From what we know of this order, however,
there seems also no basis for this assumption. W e have,
therefore, to confess that we are not certain that the
numerals were alphabetic at all, and if they were alpha-
betic we have no evidence at present as to the basis of
selection. The later forms may possibly have been alpha-
betical expressions of certain syllables called aksaras,
which possessed in Sanskrit fixed numerical values,2 but
this is equally uncertain with the rest. Bay ley also
thought3 that some of the forms were Phoenician, as
notably the use of a circle for twenty, but the resem-
blance is in general too remote to be convincing.

There is also some slight possibility that Chinese influ-
ence is to be seen hi certain of the early forms of Hindu
numerals.4

1 For a general discussion of the connection between the numerals
and the different kinds of alphabets, see the articles by U. Ceretti,
“Sulla origins delle cifre numerali moderne,” Rivista difisica, mate-
matica e scienze naturali , Pisa and Pavia, 1909, annoX, numbers 114,
118, 119, and 120, and continuation in 1910.

2 This is one of Biihler’s hypotheses. See Bayley, loc. cit., reprint
p. 4 ; a good bibliography of original sources is given in this work, p. 38.

3 Loc. cit., reprint, part I, pp. 12, 17. See also Burnell, loc. cit.,
p. 04, and tables in plate XXIII.

4 This was asserted by G. Hager ( Memoria sulle cifre arabiche,
Milan, 1813, also published in Fundgruben des Orients , Vienna, 1811,
and in BibliotMque Britannique , Geneva, 1812). See also the recent
article by Major Charles E. Woodruff, “The Evolution of Modern
Numerals from Tally Marks,” American Mathematical Monthly , August-
September, 1909. Biernatzki, “ Die Aritlimetik der Chinesen,” Crelle's
Journal fur die reine und angewandte Mathematik , Vol. LII, 1857,
pp. 59-90, also asserts the priority of the Chinese claim for a place
system and the zero, but upon the flimsiest authority. Ch. de Para-
vey, Essai sur Vorigine unique et hidroglyphique des chiffres et des lettres
de tous les peuples , Paris, 1820; G. Kleinwachter, “The Origin of the
Arabic Numerals,” China Review , Vol. XI, 1882-1883, pp. 379-381,
Vol. XII, pp. 28-30; Biot, “Note sur la connaissance que les Chinois
out eue de la valeur de position des chiffres,” Journal Asiatique, 1839,

EARLY HINDU FORMS WITH NO PLACE VALUE 33

More absurd is the hypothesis of a Greek origin, sup-
posedly supported by derivation of the current symbols
from the first nine letters of the Greek alphabet.1 This
difficult feat is accomplished by twisting some of the
letters, cutting off, adding on, and effecting other changes
to make the letters fit the theory. This peculiar theory
was first set up by Dasypodius 2 (Conrad Rauhfuss), and
was later elaborated by Huet.3

pp. 497-502. A. Terrien de Lacouperie, “The Old Numerals, the
Counting-Rods and the Swan-Pan in China,” Numismatic Chronicle ,
Vol. Ill (3), pp. 297-340, and Crowder B. Moseley, “Numeral Char-
acters: Theory of Origin and Development,” American Antiquarian ,
Vol. XXII, pp. 279-284, both propose to derive our numerals from
Chinese characters, in much the same way as is done by Major Wood-
ruff, in the article above cited.

1 The Greeks, probably following the Semitic custom, used nine
letters of the alphabet for the numerals from 1 to 9, then nine others
for 10 to 90, and further letters to represent 100 to 900. As the ordi-
nary Greek alphabet was insufficient, containing only twenty-four
letters, an alphabet of twenty-seven letters was used.

2 Institutions mathematicae , 2 vols., Strassburg, 1593-1596, a some-
what rare work from which the following quotation is taken :

“ Quis est harum Cyphrarum autor ?

“ A quibus hae usitatae syphrarum notae sint inventae : hactenus
incertum fuit : meo tamen iudicio, quod exiguum esse fateor : a grae-
cis librarijs (quorum olim magna fuit copia) literae Graecorum quibus
veteres Graeci tamquam numerorum notis sunt usi : fuerunt corruptae.
vt ex his licet videre.

“ Graecorum Literae corruptae.

narij, numeri nota, nostrae notae, quibus hodie utimur: ab his sola
differunt elegantia, vt apparet.”

See also Bayer, Historia regni Graecorum Bactriani, St. Petersburg,
1738, pp. 129-130, quoted by Martin, Recherches nouvelles , etc., loc. cit.

8 P. D. Huet, Demonstratio evangelica, Paris, 1769, note to p. 139 on
p. 647 : “Ab Arabibus vel ab Indis inventas esse, non vulgus eruditorum

* (3r/t57 ^
I / / y < v j
I (olX°)

“ Sed qua ratione graecorum
literae ita fuerunt corruptae ?

“ Finxerunt has corruptas
Graecorum literarum notas : vel
abiectione vt in nota binarij nu-
meri, vel additione vt in terna-
rij, vel inuersione vt in septe-

THE IIINDU-ARABIC NUMERALS

A bizarre derivation based upon early Arabic (c. 1040
a.d.) sources is given by Kircher in his work 1 on number
mysticism. He quotes from Abenragel,2 giving the Ara-
bic and a Latin translation 3 and stating that the ordinary
Arabic forms are derived from sectors of a circle, ©.

Out of all these conflicting theories, and from all the
resemblances seen or imagined between the numerals of
the W est and those of the East, what conclusions are we
prepared to draw as the evidence now stands ? Probably
none that is satisfactory. Indeed, upon the evidence at

modo, sed doctissimi quique ad banc diom arbitrati sunt. Ego vero
falsum id esse, merosque esse Graecorum characteres aio ; librariis
Graecae linguae ignaris interpolates, et diuturna scribendi consuetu-
dine corruptos. Nam primum i apex fuit, seu virgula, nota fj.ova.oos. z,
est ipsum p extremis suis truncatum. y, si in sinistram partem incli-
naveris & cauda mutilaveris & sinistrum cornu sinistrorsum flexeris,
fiet 3. Res ipsa loquitur 4 ipsissimum esse A, cujus crus sinistrum
erigitur tear a naderov, & infra basim descendit; basis vero ipsa ultra
crus producta eminet. Vides quam 5 simile sit ry infimo tantum
semicirculo, qui sinistrorsum patebat, dextrorsum converso. iirla-ripov
pad quod ita notabatur g-, rotundato ventre, pede detracto, peperit to 6.
Ex Z basi sua mutilato, ortum est to 7. Si H inflexis introrsum api-
cibus in rotundiorem & commodiorem formam mutaveris, exurget t6 8.
At 9 ipsissimum est

I. Weidler, Spicilegium obsemationum ad historiam notarum nu-
meralium , Wittenberg, 1755, derives them from the Hebrew letters;
Dom Augustin Calmet, “Recherches sur l’origine des clhffres d’aritli-
mdtique,” M 6-moires pour Vhistoire des sciences et des beaux arts , Tre-
voux, 1707 (pp. 1G20-1G35, with two plates), derives the current symbols
from the Romans, stating that they are relics of the ancient “ Notae
Tironianae.” These “ notes ” were part of a system of shorthand in-
vented, or at least perfected, by Tiro, a slave who'was freed by Cicero.
L. A. Sedillot, “Sur l’origine de nos chiffres,” Atti dell' Accademia
pontificia dei nuovi Lined , Vol. XVIII, 1864-18G5, pp. 316-322, derives
the Arabic forms fi’om the Roman numerals.

1 Athanasius Kircher, Arithmologia sive De abditis Numerorum
mysterijs qua origo , antiquitas dt fabrica Numerorum exponitur, Rome,
1G65.

2 See Suter, Die Mathemaliker und Astronomen der Araber, p. 100.

8 “ Et hi numeri sunt numeri Indiani, a Brachmanis Indiae Sapi-

entibus ex figura circuli secti inuenti,”

EARLY HINDU FORMS WITH NO PLACE VALUE 35

hand we might properly feel that everything points to
the numerals as being substantially indigenous to India.
And why should this not be the case ? If the king
Srong-tsan-Gampo (639 A.D.), the founder of Lhasa,1
could have set about to devise a new alphabet for Tibet,
and if the Siamese, and the Singhalese, and the Burmese,
and other peoples in the East, could have created alpha-
bets of their own, why should not the numerals also have
been fashioned by some temple school, or some king, or
some merchant guild ? By way of illustration, there are
shown in the table on page 36 certain systems of the
East, and while a few resemblances are evident, it is
also evident that the creators of each system endeavored
to find original forms that should not be found in other
systems. This, then, would seem to be a fair interpreta-
tion of the evidence. A human mind cannot readily
create simple forms that are absolutely new ; what it
fashions will naturally resemble what other minds have
fashioned, or what it has known through hearsay or
through sight. A circle is one- of the world’s common
stock of figures, and that it should mean twenty in Phoe-
nicia and in India is hardly more surprising than that
it signified ten at one time in Babylon.2 It is therefore
quite probable that an extraneous origin cannot be found
for the very sufficient reason that none exists.

Of absolute nonsense about the origin of the sym-
bols which we use much has been written. Conjectures,

1 V. A. Smith, The Early History of India , Oxford, 2d ed., 1908,
p. 333.

2 C. J. Ball, “An Inscribed Limestone Tablet from Sippara,” Pro-
ceedings of the Society of Biblical Archaeology , Vol. XX, p. 25 (Lon-
don, 1898). Terrien de Lacouperie states that the Chinese used the
circle for 10 before the beginning of the Christian era. [ Catalogue of
Chinese Coins , London, 1892, p. xl.]

86 THE IIINHU-ARABIC NUMERALS

however, without any historical evidence for support,
have no place in a serious discussion of the gradual evo-
lution of the present numeral forms.1 2 3 4 * 6

Table of Certain Eastern Systems

Siam

TfD oi (s <? b

s Burma

o 9

J ? 7 Q © <?ra e

3Malabar

O Of

A- g) y rj ^

o aJ

* Tibet

(JOT

B Ceylon

8 Malayalam

CL "L & O "3 rrt

1 For a purely fanciful derivation from the corresponding number
of strokes, see W. W. R. Ball, A SJiort Account of the History of Mathe-
matics, 1st ed., London, 1888, p. 147 ; similarly J. B. Reveillaud, Essai
sur les chiffres arabes , Paris, 1883; P. Voizot, “Les chiffres arabes et
leur origine,” La Nature , 1899, p. 222 ; G. Dumesnil, “De la forme des
chiffres usuels,” Annales de V university de Grenoble , 1907, Yol. XIX,
pp. G57-G74, also a note in Revue Archdologique , 1890, Yol. XVI (3),
pp. 342-348 ; one of the earliest references to a possible derivation
from points is in a work by Bettino entitled Apiaria universae pliilo-
sopliiae mathematicae in quibus paradoxa et noua machinamenta ad usus
eximios traducta , et facillimis demonstrationibus confirmata, Bologna,
1545, Vol. II, Apiarium XI, p. 5.

2 Alphabetum Barmanum , Romae, mdcclxxvi, p. 50. The 1 is evi-
dently Sanskrit, and the 4, 7, and possibly 9 are from India.

3 Alphabetum Grandonico-Malabaricum , Romae, mdcclxxii, p. 90.
The zero is not used, but the symbols for 10, 100, and so on, are joined
to the units to make the higher numbers.

4 Alphabetum Tangidanum , Romae, mdcclxxiii, p. 107. In a Ti-

betan MS. in the library of Professor Smith, probably of the eigh-
teenth century, substantially these forms are given.

6 Bayley, loc. cit., plate II. Similar forms to these here shown, and
numerous other forms found in India, as well as those of other oriental
countries, are given by A. P. Pihan, Expose des signes de numeration
ysites chez lespeuples orientaux ancicns et modernes, Paris, 1860.

EARLY HINDU FORMS WITH NO PLACE VALUE 37

We may summarize this chapter by saying that no one
knows what suggested certain of the early numeral forms
used in India. The origin of some is evident, but the
origin of others will probably never be known. There is
no reason why they should not have been invented by
some priest or teacher or guild, by the order of some
king, or as part of the mysticism of some temple. What-
ever the origin, they were no better than scores of other
ancient systems and no better than the present Chinese
system when written without the zero, and there would
never have been any chance of their triumphal progress
westward had it not been for this relatively late symbol.
There could hardly be demanded a stronger proof of the
Hindu origin of the character for zero than this, and to
it further reference will be made in Chapter IV.

CHAPTER III

LATER HINDU FORMS, WITH A PLACE VALUE

Before speaking of the perfected Hindu numerals with
the zero and the place value, it is necessary to consider
the third system mentioned on page 19, — the word and
letter forms. The use of words with place value began
at least as early as the 6th century of the Christian era.
In many of the manuals of astronomy and mathematics,
and often in other works in mentioning dates, numbers
are represented by the names of certain objects or ideas.
For example, zero is represented by “the void” (suny a),
or “ heaven-space ” (ambara akasa ) ; one by “ stick ”
(rupa), “ moon ” (indu Sami'), “ earth ” (bhu), “ begin-
ning ” ( adi ), “ Brahma,” or, in general, by anything
markedly unique ; two by “ the twins ” (yama), “ hands ”
(kara), “ eyes ” (nayana), etc. ; four by “ oceans,” five
by “ senses ” (visaya) or “ arrows ” (the five arrows of
Kamadeva) ; six by “ seasons ” or “ flavors ” ; seven by
“ mountain ” (ago), and so on.1 These names, accommo-
dating themselves to the verse in which scientific works
were written, had the additional advantage of not admit-
ting, as did the figures, easy alteration, since any change
would tend to disturb the meter.

1 Biihler, loc. cit., p. 80; J. F. Fleet, Corpus inscriptionum, Indica-
rum , Vol. Ill, Calcutta, 1888. Lists of such words are given also by
Al-Biruni in his work India-, by Burnell, loc. cit.; by E. Jacquet,
“ Mode d’ expression symbolique des nombres employ^ par les Indians,
lesTib6tains et les Javanais,” Journal Asiatique, Vol. XVI, Paris, 1835.

LATER HINDU FORMS WITH A PLACE VALUE 39

As an example of this system, the date “ Saka Samvat,
867” (a.d. 945 or 946), is given by u giri-ram-vasu,”
meaning “ the mountains ” (seven), “ the flavors ” (six),
and the gods “ Vasu ” of which there were eight. In read-
ing the date these are read from right to left.1 The
period of invention of this system is uncertain. The first
trace seems to be in the Srautasutra of Katyayana and
Latyayana.2 It was certainly known to Varaha-Mihira
(d. 587), 3 for he used it in the Brhat-Samhita. 4 It has
also been asserted5 that Aryabhata (c. 500 A.D.) was
familiar with this system, but there is nothing to prove
the statement.6 The earliest epigraphical examples of
the system are found in the Bayang (Cambodia) inscrip-
tions of 604 and 624 a.d.7

Mention should also be made, in this connection, of a
curious system of alphabetic numerals that sprang up in
southern India. In this we have the numerals repre-
sented by the letters as given in the following table :

1 2

G 7

8 9 0

k kh

c ch

3 jh n

t tli

dli

t th

d dli n

P Ph

y r

s s

h 1

1 This date is given by Fleet, loc. cit., Vol. Ill, p. 73, as the earliest

epigraphical instance of this usage in India proper.

2 Weber, Indische Studien, Vol. VIII, p. 166 seq.

3 Journal of the Royal Asiatic Society , Vol. I (n.s.), p. 407.

4 VIII, 20, 21.

6 Th. H. Martin, Les sicjnes numdraux . . ., Rome, 1864; Lassen,
Indische Alterthumskunde,Vo\. II, 2d ed., Leipzig and London, 1874,
p. 1153.

6 But see Burnell, loc. cit., and Thibaut, Astronomie, Astrologie und
Mathematik, p. 71.

7 A.. Barth, “ Inscriptions Sanscrites du Cambodge,” in the Notices et
extraits des Mss. de la BibliotMque nationale, Vol. XXVII, Part I, pp. 1-
180, 1885; see also numerous articles in Journal Asiatique, byAymonier.

THE IiINDU-ARABIC NUMERALS

By this plan a numeral might be represented by any
one of several letters, as shown in the preceding table,
and thus it could the more easily be formed into a word
for mnemonic purposes. For example, the word

2 3 1 5 6 5 1

kka gont yan me sa ma pa

has the value 1,565,132, reading from right to left.1 This,
the oldest specimen (1184 a.d.) known of this notation,
is given in a commentary on the Rigveda, representing
the number of days that had elapsed from the beginning
of the Kaliyuga. Burnell 2 states that this system is
even yet in use for remembering rules to calculate horo-
scopes, and for astronomical tables.

A second system of this kind is still used in the
pagination of manuscripts in Ceylon, Siam, and Burma,
havino’ also had its rise hi southern India. In this the

thirty -four consonants when followed by a (as ka . . . let)
designate the numbers 1-34 ; by d (as ka . . . /a), those
from 35 to 68 ; by i (hi . . . li ), those from 69 to 102,
inclusive ; and so on.3

As already stated, however, the Hindu system as thus
far described was no improvement upon many others of
the ancients, such as those used by the Greeks and the
Hebrews. Having no zero, it was impracticable to desig-
nate the tens, hundreds, and other units of higher order
by the same symbols used for the units from one to nine.
In other words, there was no possibility of place value
without some further improvement. So the Nana Ghat

1 Biihler, loc. cit., p. 82.

2 Loc. cit., p. 79.

3 Biihler, loc. cit., p. 83. The Hindu astrologers still use an alpha-
betical system of numerals. [Burnell, loc. cit., p. 79.]

LATER HINDU FORMS WITH A PLACE VALUE 41

symbols required the writing of “ thousand seven twenty-
four ” about like T 7, tw, 4 in modern symbols, instead
of 7024, in which the seven of the thousands, the two
of the tens (concealed in the word twenty, being origi-
nally “twain of tens,” the -ty signifying ten), and the
four of the units are given as spoken and the order of
the unit (tens, hundreds, etc.) is given by the place. To
complete the system only the zero was needed ; but it
was probably eight centuries after the Nana Ghat inscrip-
tions were cut, before this important symbol appeared ;
and not until a considerably later period did it become
well known. Who it was to whom the invention is due,
or where he lived, or even in what century, will probably
always remain a mystery.1 It is possible that one of the
forms of ancient abacus suggested to some Hindu astron-
omer or mathematician the use of a symbol to stand for
the vacant line when the counters were removed. It is
well established that in different parts of India the names
of the higher powers took different forms, even the order
being interchanged.2 Nevertheless, as the significance of
the name of the unit was given by the order in reading,
these variations did not lead to error. Indeed the varia-
tion itself may have necessitated the introduction of a
word to signify a vacant place or lacking unit, with the
ultimate introduction of a zero symbol for tins word.

To enable us to appreciate the force of this argument
a large number, 8,443,682,155, may be considered as the
Hindus wrote and read it, and then, by way of contrast,
as the Greeks and Arabs would have read it.

1 Well could Ramus say, “Quicunq; autem fuerit inventor decern
notarum laudem magnam meruit.”

2 Al-BirunI gives lists.

THE IIINDU-ARABIC NUMERALS

Modem American reading , 8 billion, 443 million, 682
thousand, 155.

Hindu, 8 padmas, 4 vyarbudas, 4 kotis, 3 prayutas,
6 laksas, 8 ayutas, 2 sahasra, 1 sata, 5 dasan, 5.

Arabic and early German, eight thousand thousand
thousand and four hundred thousand thousand and forty-
three thousand thousand, and six hundred thousand and
eighty-two thousand and one hundred fifty -five (or five
and fifty).

Greek, eighty-four myriads of myriads and four thou-
sand three hundred sixty-eight myriads and two thou-
sand and one hundred fifty-five.

As W oepcke 1 pointed out, the reading of numbers of
this kind shows that the notation adopted by the Hindus
tended to bring out the place idea. No other language
than the Sanskrit has made such consistent application,
in numeration, of the decimal system of numbers. The
introduction of myriads as in the Greek, and thousands
as in Arabic and in modern numeration, is really a step
away from a decimal scheme. So in the numbers below
one hundred, in English, eleven and twelve are out of
harmony with the rest of the -teens, while the naming of
all the numbers between ten and twenty is not analogous
to the naming of the numbers above twenty. To conform
to our written system we should have ten-one, ten-two,
ten-three, and so on, as we have twenty-one, twenty-two,
and the like. The Sanskrit is consistent, the units, how-
ever, preceding the tens and hundreds. Nor did any
other ancient people carry the numeration as far as did
the Hindus.2

1 Propagation, loc. cit., p. 443.

2 See the quotation from The Light of Asia in Chapter II, p. 10.

LATER HINDU FORMS WITH A PLACE VALUE 43

When the anJcapalli,1 the decimal-place system of writ-
ing numbers, was perfected, the tenth symbol was called
the sunyabindu, generally shortened to siinya (the void).
Brockhaus 2 has well said that if there was any invention
for which the Hindus, by all their philosophy and reli-
gion, were well fitted, it was the invention of a symbol
for zero. This making of nothingness the crux of a tre-
mendous achievement was a step in complete harmony
with the genius of the Hindu.

It is generally thought that this siinya as a symbol
was not used before about 500 a.d., although some writ-
ers have placed it earlier.3 Since Aryabhata gives our
common method of extracting roots, it would seem that
he may have known a decimal notation,4 although he
did not use the characters from which our numerals
are derived.6 Moreover, he frequently speaks of the

1 The nine ciphers were called aiika.

2 “ Zur Geschichte des indischen Ziffernsystems,” Zeitschrift fur die
Kunde des Morgenlandes, Vol. IV, 1842, pp. 74-83.

3 It is found in the Bakhsall MS. of an elementary arithmetic
which Hoernle placed, at first, about the beginning of our era, but the
date is much in question. G. Tliibaut, loc. cit., places it between 700
and 900 a.d. ; Cantor places the body of the work about the third or
fourth century a.d., Geschichte der Mathematik , Vol. I (3), p. 598.

4 For the opposite side of the case see G. R. Kaye, “ Notes on Indian
Mathematics, No. 2. — Aryabhata,” Journ. and Proc. of the Asiatic Soc.
of Bengal, Vol. IV, 1908, pp. 111-141.

6 lie used one of the alphabetic systems explained above. This ran
up to 1018 and was not difficult, beginning as follows :

the same letter (lea) appearing in the successive consonant forms, ka,
kha, ga , glia, etc. See C. I. Gerhardt, tiber die Entsteliung und Aus-
breitung des dekadisclien Zahlensy stems, Programm, p. 17, Salzwedel,
1853, and Etudes historiques sur V arithmetique de position, Programm,
p. 24, Berlin, 1856; E. Jacquet, Mode d' expression symbolique des nombres,

l 102 iQ4 ioo los, etc.,

TIIE IIINDU-ARABIC NUMERALS

void.1 If he refers to a symbol this would put the zero
as far back as 500 a.d., but of course he may have re-
ferred merely to the concept of nothingness.

A little later, but also in the sixth century, Varaha-
Mihira2 wrote a work entitled Brhat /Samhita 3 in which
he frequently uses sunya in speaking of numerals, so
that it has been thought that lie was referring to a defi-
nite symbol. This, of course, would add to the proba-
bility that Aryabhata was doing the same.

It should also be mentioned as a matter of interest, and
somewhat related to the question at issue, that V araha-
Mihira used the word-system with place value 4 as ex-
plained above.

The first kind of alphabetic numerals and also the
word-system (in both of which the place value is used)
are plays upon, or variations of, position arithmetic, which
would be most likely to occur in the country of its origin.5

At the opening of the next century (c. 620 a.d.) Bana6
wrote of Subandhus’s Vasavadatta as a celebrated work,

loc. cit., p. 97 ; L. Rodet, Sur la veritable signification de la notation
numdrique inventde par Aryabhata,” Journal Asiatique, Vol. XVI (7),
pp. 440-485. On the two Aryabhatas see Kaye, Bibl. Math., Vol. X (3),
p. 289.

1 Using kha, a synonym of sunya. [Bayley, loc. cit., p. 22, and L.
Rodet, Journal Asiatique, Vol. XVI (7), p. 443.]

2 Varaha-Miliira, Pahcasiddhdntika , translated by G. Thibaut and
M. S. Dvivedi, Benares, 1889; see Biihler, loc. cit., p. 78; Bayley,
loc. cit., p. 23.

8 Brhat Samhita, translated by Kern, Journal of the Royal Asiatic
Society, 1870-i875.

4 It is stated by Biihler in a personal letter to Bayley (loc. cit., p. G5)
that there are hundreds of instances of this usage in the Brhat Sam-
hita. The system was also used in the Pahcasiddhdntika as early as
505 a.d. [Biihler, Palaeoqraphie, p. 80, and Fleet, Journal of the Royal
Asiatic Society, 1910, p. 819.]

6 Cantor, Geschichte der Mathematik, Vol. I (3), p. 008.

G Biihler, loc. cit., p. 78.

LATER HINDU FORMS WITH A PLACE VALUE 45

and mentioned that the stars dotting the sky are here
compared with zeros, these being points as in the mod-
ern Arabic system. On the other hand, a strong argu-
ment against any Hindu knowledge of the symbol zero
at this time is the fact that about 700 a.d. the Arabs
overran the province of Sind and thus had an opportu-
nity of knowing the common methods used there for
writing numbers. And yet, when they received the com-
plete system in 776 they looked upon it as something
new.1 Such evidence is not conclusive, but it tends to
show that the complete system was probably not in com-
mon use in India at the beginning of the eighth century.
On the other hand, we must bear in mind the fact that
a traveler in Germany in the year 1700 would probably
have heard or seen nothing of decimal fractions, although
these were perfected a century before that date. The
elite of the mathematicians may have known the zero
even in Aryabhata’s time, while the merchants and the
common people may not have grasped the significance of
the novelty until a long time after. On the whole, the
evidence seems to point to the west coast of India as the
region where the complete system was first seen.2 As
mentioned above, traces of the numeral words with place
value, which do not, however, absolutely require a deci-
mal place-system of symbols, are found very early in
Cambodia, as well as in India.

Concerning the earliest epigraphical instances of the use
of the nine symbols, plus the zero, with place value, there

1 Bayley, p. 38.

2 Novioinagus, in his Be numeris libri duo , Paris, 1539, confesses his
ignorance as to the origin of the zero, but says : “ D. Henricus Grauius,
vir Graecfe & Hebraic^ exiled doctus, Iiebraicam originem ostendit,”
adding that Valla “Indis Orieutalibus gentibus inventionem tribuit.”

THE IIINDU-ARABIC NUMERALS

is some question. Colebrooke 1 in 1807 warned against
the possibility of forgery in many of the ancient copper-
plate land grants. On this account Fleet, in the Indian
Antiquary ,2 discusses at length this phase of the work of
the epigraphists in India, holding that many of these
forgeries were made about the end of the eleventh cen-
tury. Colebrooke 3 takes a more rational view of these
forgeries than does Kaye, who seems to hold that they
tend to invalidate the whole Indian hypothesis. “But
even where that may be suspected, the historical uses of
a monument fabricated so much nearer to the times to
which it assumes to belong, will not be entirely super-
seded. The necessity of rendering the forged grant credi-
ble would compel a fabricator to adhere to history, and
conform to established notions : and the tradition, which
prevailed in his time, and by which he must be guided,
would probably be so much nearer to the truth, as it
was less remote from the period which it concerned.” 4
Bidder6 gives the copper-plate Gurjara inscription of
Cedi-samvat 346 (595 a.d.) as the oldest epigraphical
use of the numerals 6 “ in which the symbols correspond
to the alphabet numerals of the period and the place.”
Vincent A. Smith 7 quotes a stone inscription of 815 a.d.,
dated Samvat 872. So F. Ivielhorn in the Epigraphia
Indica 8 gives a Pathari pillar inscription of Parabala,
dated Vikrama-samvat 91 7, which corresponds to 861 A. D.,

1 See Essays, Vol.. II, pp. 287 and 288.

2 Vol. XXX, p. 205 seqq. 8 Loc. cit., p. 284 seqq.

4 Colebrooke, loc. cit., p. 288. 5 Loc. cit., p. 78.

6 Hereafter, unless expressly stated to the contrary, we shall use
the word “ numerals” to mean numerals with place value.

v “The Gurjaras of Rajputana and Kanauj,” in Journal of the Royal
Asiatic Society, January and April, 1909.

8 Vol. IX, 1908, p. 248.

LATER HINDU FORMS WITH A PLACE VALUE 47

and refers also to another copper-plate inscription dated
V ikrama-samvat 813 (756 a.d.). The inscription quoted
by V. A. Smith above is that given by D. R. Bhan-
darkar,1 and another is given by the same writer as of
date Saka-samvat 715 (798 a.d.), being incised on a
pilaster. Kielhorn2 also gives two copper-plate inscrip-
tions of the time of Mahendrapala of Kanauj, Valhabl-
samvat 574 (893 a.d.) and Vikrama-samvat 956 (899
a.d.). That there should be any inscriptions of date as
early even as 750 a.d., would tend to show that the sys-
tem was at least a century older. As will be shown in
the further development, it was more than two centu-
ries after the introduction of the numerals into Europe
that they appeared there upon coins and inscriptions.
While Thibaut 3 does not consider it necessary to quote
any specific instances of the use of the numerals, he
states that traces are found from 590 a.d. on. “ That
the system now in use by all civilized nations is of Hindu
origin cannot be doubted ; no other nation has any claim
upon its discovery, especially since the references to the
origin of the system which are found in the nations of
western Asia point unanimously towards India.” 4

The testimony and opinions of men like Bidder, Kiel-
horn, V. A. Smith, Bhandarkar, and Thibaut are entitled
to the most serious consideration. As authorities on
ancient Indian epigraphy no others rank higher. Their
work is accepted by Indian scholars the world over, and
their united judgment as to the rise of the system with
a place value — that it took place in India as early as the

1 Epigraphia Indica, Vol. IX, pp. 193 and 198.

2 Epigraphia Indica , Vol. IX, p. 1.

8 Loc. cit., p. 71. 4 Thibaut, p. 71.

THE HINDU- A R A BIG NUMERALS

sixth century a.d. — must stand unless new evidence of
great weight can be submitted to the contrary.

Many early writers remarked upon the diversity of
Indian numeral forms. Al-B Irani was probably the first ;
noteworthy is also Johannes Hispalensis,1 who gives the
variant forms for seven and four. We insert on p. 49 a
table of numerals used with place value. While the chief
authority for this is Bidder,2 several specimens are given
which are not found in his work and which are of unusual
interest.

»*

The Sarada forms given in the table use the circle as a
symbol for 1 and the dot for zero. They are taken from
the paging and text of The Kashmirian Atharva-Veda ,3
of which the manuscript used is certainly four hundred
years old. Similar forms are found in a manuscript be-
longing to the University of Tubingen. Two other series
presented are from Tibetan books in the library of one
of the authors.

For purposes of comparison the modern Sanskrit and
Arabic numeral forms are added.

1 “ Est autem in aliquibus figurarum istarum apud multos diuersi-
tas. Quidam enim septiinam banc figuram representant,” etc. [Bon-
compagni, Trattati, p. 28.] Enestrom lias shown that very likely this
work is incorrectly attributed to Johannes Hispalensis. [ bibliotheca
Mathematica , Yol. IX (3), p. 2.]

2 Indische Palaeographze, Tafel IX.

3 Edited by Bloomfield and Garbe, Baltimore, 1901, containing
photographic reproductions of the manuscript.

Sanskrit,

Arabic,

LATER HINDU FORMS WITH A PLACE VALUE 49

Numerals used with Place Value

1 2 3 4 5 G 7 890

• ^ £{

Sr i y ^ °

3. Sr/l ^r<y

"7 cv a.t) 9 °

^ n 1 Jr i °

"/ <2- 3 4 J ^ 7 -V 3 °

t 'Z % £ <! 6, >> S ^ °

’/ z ^ f n s 9

‘i i: i Ky ^ °

' 0 /£ o

O $ ? * * 'h'ls? •

a Bakhsall MS. See page 43; Hoernle, R., The Indian Antiquary,
Vol. XVII, pp. 33-48, 1 plate ; Hoernle, Verhandlungen des VII. Inter-
nationalen Orientalisten-Concjr esses, Arische Section, Vienna, 1888, “On
the Bakshall Manuscript,” pp. 127-147, 3 plates ; Biihler, ioc. cit.

b 3,4,0, from II. II. Dhruva, “Three Land-Grants from Sankheda,”
Epigraphia Indica, Vol. II, pp. 19-24 with plates ; date 595 a.d. 7, 1, 5,

TIIE IIINDU-ARABIC NUMERALS

from Bhandarkar, “ Daulatabacl Plates,” Epigraphia Indica, Vol. IX,
part V ; date c. 71)8 a.d.

c 8, 7, 2, from “Buckhala Inscription of Nagabliatta,” Bhandarkar,
Epigraphia Indica, Vol. IX, part V ; date 815 a.d. 5 from “The Morbi
Copper-Plate,” Bhandarkar, The Indian Antiquary , Vol. II, pp. 257-
258, with plate ; date 804 a.d. See Btililer, loc. cit.

d 8 from the above Morbi Copper-Plate. 4, 5, 7, 9, and 0, from “Asni
Inscription of Maliipala,” The Indian Antiquary, Vol. XVI, pp. 174-
175; inscription is on red sandstone, date 917 a.d. See Biihler.

c 8, 9, 4, from “ ltaslitrakuta Grant of Amoghavarsha,” J. F. Fleet,
The Indian Antiquary, Vol. XII, pp. 263-272; copper-plate grant of
date c. 972 a.d. See Biihler. 7, 3, 5, from “Torkhede Copper-Plate
Grant of the Time of Govindaraja of Gujerat,” Fleet, Epigraphia In-
dica, Vol. Ill, pp. 53-58. See Biihler.

t From “ A Copper-Plate Grant of King Tritochanapala Clianlukya
of Latadesa,” H. H. Dhruva, Indian Antiquary, Vol. XII, pp. 196-
205; date 1050 a.d. See Biihler.

s Burnell, A. C., South Indian Paleography , plate XXIII, Telugu-
Canarese numerals of the eleventh century. See Biihler.

h and * From a manuscript of the second half of the thirteenth
century, reproduced in “ Della vita e delle opere di Leonardo Pisano,”
Baldassare Boncompagni, Rome, 1852, in Atti delV Accademia Pontificia
dei nuovi Lincei, anno V.

i and k From a fourteenth-century manuscript, as reproduced in
Della vita etc., Boncompagni, loc. cit.

1 From a Tibetan MS. in the library of D. E. Smith.
m From a Tibetan block-book in the library of D. E. Smith.
n Sarada numerals from The Kashmirian Atharva -Veda, reproduced
by chromophotography from the manuscript in .the University Library
at Tubingen, Bloomfield and Garbe, Baltimore, 1901. Somewhat sim-
ilar forms are given under “Numeration Cacliemirienne,” by Pilian,
Expos6 etc., p. 84.

CHAPTER IV

THE SYMBOL ZERO

What lias been said of the improved Hindu system
with a place value does not touch directly the origin of
a symbol for zero, although it assumes that such a sym-
bol exists. The importance of such a sign, the fact that it
is a prerequisite to a place-value system, and the further
fact that without it the Hindu-Arabic numerals would
never have dominated the computation system of the
western world, make it proper to devote a chapter to its
origin and history.

It was some centuries after the primitive Brahmi and
Kharostlii numerals had made then" appearance in India
that the zero first appeared there, although such a char-
acter was used by the Babylonians 1 in the centuries
immediately preceding the Christian era. The symbol is
£ or and apparently it was not used in calculation.
Nor does it always occur when units of any order are
lacking; thus 180 is written y Y Y with the meaning
three sixties and no units, since 181 immediately follow-
ing is YYY Y, three sixties and one unit.2 The main

1 Franz X. Kugler, Die Babylonische Mondreclinimcj , Freiburg i. Br.,
1900, in the numerous plates at the end of the book; practically all
of these contain the symbol to which reference is made. Cantor,
Geschichte , Vol. I, p. 31.

2 F. X. Kugler, Sternkunde und Sterndienst in Babel , I. Buch, from
the beginnings to the time of Christ, Munster i. Westfalen, 1907. It
also has numerous tables containing the above zero.

THE HINDU-ARABIC NUMERALS

use of this Babylonian symbol seems to have been in the
fractions, GOths, 3600ths, etc., and somewhat similar to
the Greek use of o, for ov8ev, with the meaning vacant.

“The earliest undoubted occurrence of a zero in India is
an inscription at Gwalior, dated Sam vat 933 (876 a.d.).
Where 50 garlands are mentioned (line 20), 50 is written
£j o. 270 (line 4) is written \r]o.'n 1 The Bakhsall Manu-
script 2 probably antedates this, using the point or dot as
a zero symbol. Bayley mentions a grant of Jaika Rash-
trakuta of Bharuj, found at Okamandel, of date 738 a.d.,
which contains a zero, and also a coin with indistinct
Gupta date 707 (897 a.d.), but the reliability of Bay-
ley’s work is questioned. As has been noted, the appear-
ance of the numerals in inscriptions and on coins would
be of much later occurrence than the origin and written
exposition of the system. From the period mentioned
the spread was rapid over all of India, save the southern
part, where the Tamil and Malay alam people retain the
old system even to the present day.3

Aside from its appearance in early inscriptions, there
is still another indication of the Hindu origin of the sym-
bol in the special treatment of the concept zero in the
early works on arithmetic. Brahmagupta, who lived in
Ujjain, the center of Indian astronomy,4 in the early part

1 From a letter to D. E. Smith, from G. F. Hill of the British
Museum. See also his monograph “On the Early Use of Arabic Nu-
merals in Europe,” in Archacologia , Vol. LXII (1910), p. 137.

2 R. Hoernle, “The Bakshall Manuscript,” Indian Antiquary, Vol.
XVII, pp. 33-48 and 275-279, 1888 ; Thibaut, Astronomic, Astrologie
und Mathematik, p. 75; Hoernle, Verhandlungen, loc. cit., p. 132.

3 Bayley, loc. cit., Vol. XV, p. 29. Also Bendall, “On a System of
Numerals used in South India,” Journal of the Royal Asiatic Society,
1890, pp. 789-792.

4 V. A. Smith, The Early History of India, 2d ed., Oxford, 1908,
p. 14.

THE SYMBOL ZERO

of the seventh century, gives in his arithmetic 1 a distinct
treatment of the properties of zero. He does not discuss
a symbol, but he shows by his treatment that hi some
way zero had acquired a special significance not found in
the Greek or other ancient arithmetics. A still more
scientific treatment is given by Bhaskara,2 although in
one place he permits himself an unallowed liberty in
dividing by zero. The most recently discovered work
of ancient Indian mathematical lore, the Ganita-Sara-
Sangraha3 of Mahavlracarya (c. 830 a.d.), while it does
not use the numerals with place value, has a similar dis-
cussion of the calculation with zero.

What suggested the form for the zero is, of course,
purely a matter of conjecture. The dot, which the Hin-
dus used to fill up lacunae in then manuscripts, much as
we indicate a break in a sentence,4 would have been a
more natural symbol ; and this is the one which the Hin-
dus first used 5 and which most Arabs use to-day. There
was also used for this purpose a cross, like our X, and this
is occasionally found as a zero symbol.0 In the Bakhsali
manuscript above mentioned, the word sunya, with the
dot as its symbol, is used to denote the unknown quan-
tity, as well as to denote zero. An analogous use of the

1 Colebrooke, Algebra , with Arithmetic and Mensuration , from the
Sanskrit of Brahmegupta and Bhdscara, London, 1817, pp. 339-340.

2 Ibid., p. 138.

3 D. E. Smith, in the Bibliotheca Mathematical Vol. IX (3), pp.*L06-
110.

4 As when we use three dots (...).

6 “The Hindus call the nought explicitly Sunyabindu ‘the dot
marking a blank,’ and about 500 a.d. they marked it by a simple dot,
which latter is commonly used in inscriptions and MSS. in order to
mark a blank, and which was later converted into a small circle.”
[Bidder, On the Origin of the Indian Alphabet , p. 53, note.]

0 Fazzari, Dell' origine delle parole zero e cifra , Naples, 1903.

THE IIINDU-ARABIC NUMERALS

zero, for the unknown quantity in a proportion, appears
in a Latin manuscript of some lectures by Gottfried
Wolack in the University of Erfurt in 1467 and 1468.1
The usage was noted even as early as the eighteenth
century.2

The small circle was possibly suggested by the spurred
circle which' was used for ten.3 It has also been thought
that the omicron used by Ptolemy in his Almagest , to
mark accidental blanks in the sexagesimal system which he
employed, may have influenced the Indian writers.4 This
symbol was used quite generally in Europe and Asia, and
the Arabic astronomer Al-Battani6 (died 929 a.d.) used
a similar symbol in connection with the alphabetic system
of numerals. The occasional use by Al-Battani of the
Arabic negative, la, to indicate the absence of minutes

1 E. Wappler, “Zur Geschichte der Matliematik im 15. Jalirhun-
dert,” in the Zeitschrift fur Mathematik und Physik , Yol. XLV, Ilist.-
lit. Abt., p. 47. The manuscript is No. C. 80, in the Dresden library.

2 J. G. Prandel, Algebra nebst Hirer literarischen Geschichte, p. 572,
Munich, 1795.

8 See the table, p. 23. Does the fact that the early European arith-
metics, following the Arab custom, always put the 0 after the 9, sug-
gest that the 0 was derived from the old Hindu symbol for 10 ?

4 Bayley, loc. cit., p. 48. From this fact Delambre ( Histoire de Vas-
tronomie ancienne) inferred that Ptolemy knew the zero, a theory
accepted by Chasles, Aper^u historique sur Vorigine et le ddveloppement
des mdthodes en g£om6trie, 1875 ed., p.476; Nesselmann, however, showed
( Algebra der Griechen, 1842, p. 138), that Ptolemy merely used o for
ov8£v, with no notion of zero. See also G. Eazzari, “Dell’ origine delle
parole zero e cifra,” Ateneo , Anno I, No. 11, reprinted at Naples in
1903, where the use of the point and the small cross for zero is also
mentioned. Th. H. Martin, Les signes numdraux etc., reprint p. 30, and
J. Brandis, Das Miinz-, Mass- und Gewichtswesen in Vorderasien bis auf
Alexander den Grossen, Berlin, 1866, p. 10, also discuss this usage of o,
without the notion of place value, by the Greeks.

6 Al-Battani sive Albatenii opus astronomicum. Ad fidem codicis
escurialensis arabice editum, latine versum, adnotationibus instruction
a Carolo Alphonso Nallino, 1899-1907. Publicazioni del R. Osserva-
torio di Brera in Milano, No. XL.

THE SYMBOL ZERO

(or seconds), is noted by Nallino.1 Noteworthy is also
the use of the o for unity in the &arada characters of the
Kashmirian Atharva-V eda, the writing being at least 400
years old. Bhaskara (c. 1150) used a small circle above
a number to indicate subtraction, and in the Tartar writ-
ing a redundant word is removed by drawing an oval
around it. It would be interesting to know whether our
score mark (4), read “ four in the hole,” could trace its
pedigree to the same sources. O’Creat2 (c. 1130), in a
letter to his teacher, Adelhard of Bath, uses r for zero,
being an abbreviation for the word teca which we shall
see was one of the names used for zero, although it could
quite as well be from r^typa. More rarely O’Creat uses
O, applying the name cyfra to both forms. Frater Sigs-
boto3 (c. 1150) uses the same symbol. Other peculiar
forms are noted by Heiberg 4 as being in use among the
Byzantine Greeks in the fifteenth century. It is evident
from the text that some of these writers did not under-
stand the import of the new system.5

Although the dot was used at first in India, as noted
above, the small circle later replaced it and continues in
use to tliis day. The Arabs, however, did not adopt the

1 Loc. cit., Yol. II, p. 271.

2 C. Henry, “PrologusN. Ocreati in Helceph acl Adelarclum Baten-
sem magistrum suum,” Abhandlungen zur Geschichte der Mathematik,
Vol. Ill, 1880.

3 Max. Curtze, “Ueber eine Algorismus-Schrift des XII. Jahrliun-
derts,” Abhandlungen zur Geschichte der Mathematik, Yol. VIII, 1898,
pp. 1-27 ; Alfred Nagl, “ Ueber eine Algorismus-Schrift des XII. Jahr-
hundertsund iiber die Verbreitung der indisch-arabischen Rechenkunst
und Zalilzeichen im christl. Abendlande,” Zeitschrift fur Mathematik
und Physik , Hist.-lit. Abth., Vol. XXXIV, pp. 129-146 and 161-170,
with one plate.

4 “ Byzantinische Analekten,” Abhandlungen zur Geschichte der
Mathematik , Vol. IX, pp. 161-189.

6 4 or q for 0. M also used for 5. | for 13. [Heiberg, loc. cit.]

THE IITNDU-ARABIC NUMERALS

circle, since it bore some resemblance to the letter which
expressed the number five in the alphabet system.1 The
earliest Arabic zero known is the dot, used in a manu-
script of 873 a.d.2 Sometimes both the dot and the circle
are used in the same work, having the same meaning,
which is the case in an Arabic MS., an abridged arith-
metic of Jamshid,3 982 a.h. (1575 a.d.). As given in
this work the numerals are *3 A\/^ £> f6 The form

for 5 varies, in some works becoming <p or ®; O is '
found in Egypt and appears in some fonts of type.
To-day the Arabs use the 0 only when, under European
influence, they adopt the ordinary system. Among the
Chinese the first definite trace of zero is in the work of
Tsin4 of 1247 a.d. The form is the circular one of the
Hindus, and undoubtedly was brought to China by some
traveler.

The name of this all-important symbol also demands
some attention, especially as we are even yet quite un-
decided as to what to call it. We speak of it to-day as
zero , naught , and even cipher; the telephone operator
often calls it 0, and the illiterate or careless person calls
it aught. In view of all this uncertainty we may well
inquire what it has been called in the past.5

1 Gerhardt, Etudes historiques sur V arithmetique deposition , Berlin,
1856, p. 12; J. Bowring, The Decimal System in Numbers, Coins, & Ac-
counts, London, 1854, p. 33.

2 Karabacek, Wiener Zeitschrift fiir die Kunde des Morgenlandes,
Vol. XI, p. 13 ; Fiilirer durch die Papyrus-Ausstellung Erzlierzog Rainer,
Vienna, 1894, p. 216.

3 In the library of G. A. Plimpton, Esq.

* Cantor, Geschichte , Vol. I (3), p. 674; Y. Mikami, “A Remark on
the Chinese Mathematics in Cantor’s Geschichte der Mathematik,”
Archiv der Mathematik und Physik, Vol. XV (3), pp. 68-70.

6 Of course the earlier historians made innumerable guesses as to
the origin of the word cipher. E.g. Matthew Hostus, De numeratione

TIIE SYMBOL ZERO

As already stated, the Hindus called it sunya , u void.” 1
This passed over into the Arabic as as-sifr or sifr.'2 When
Leonard of Pisa (1202) wrote upon the Hindu numerals
he spoke of this character as zephirum .3 Maximus P la-
nudes (1330), writing under both the Greek and the Ara-
bic influence, called it tziphra .4 In a treatise on arithmetic
written in the Italian language by Jacob of Florence6

emendata, Antwerp, 1582, p. 10, says: “ Siphra vox Hebrseam originem
sapit refbrtque : & ut docti arbitrantur, verbo saphar, quod Ordine
numerauit significat. Unde Sephar nulnerus est: bine Siphra (vulgo
corruptius). Etsi verb gens Iudaica his notis, qute liodie Siphrse
vocantur, usa non fuit : mansit tamen rei appellatio apud multas
gentes.” Dasypodius, Institutiones mathematicae , Yol. I, 1593, gives a
large part of this quotation word for word, without any mention of
the source. Hermannus Hugo, He prima scribendi origine, Trajecti ad
Rlienum, 1738, pp. 304-305, and note, p. 305; Karl Krumbachcr,
“Woher stammt das Wort Ziffer (Chiffre) ? ”, Etudes de philologie
nfo-grecque, Paris, 1892.

1 Buhler, loc. cit., p. 78 and p. 86.

2 Fazzari, loc. cit., p. 4. So Elia Misrachi (1455-1526) in his post-
humous Book of Number, Constantinople, 1534, explains sifra as being
Arabic. See also Steinschneider, Bibliotheca Mathematica, 1893, p. 69,
and G. Wertheim, Die Arithmetik des Elia Misrachi , Programm, Frank-
furt, 1893.

8 “Cum his novem figuris, et cum hoc signo 0, quod arabice zephi-
rum appellatur, scribitur quilibet numerus.”

4 T{l<t>pa, a form also used by Neophytos (date unknown, probably
c. 1330). It is curious that Finaeus (1555 ed., f. 2) used the form tzi-
phra throughout. A. J. H. Vincent [“Sur l’origine de nos chiffres,”
Notices et Extraits des MSS., Paris, 1847, pp. 143-150] says: “ Ce cercle
fut nornmb par les uns, sipos, rota, galgal . . . ; par les autres tsiphra
(de~l2i>‘, couronne ou diademe) ou ciphra (de "1CD, numeration).'1' Cli.
de Paravey, Essai sur V origine unique et hiAroglyphique des chiffres et des
lettres de tous les peuples, Paris, 1826, p. 165, a rather fanciful work,
gives “ vase, vase arrondi et fermd par un couvercle, qui est le symbole
de la 10° Ileure, J,” among the Chinese ; also “Tsiphron Zbron, ou
tout & fait vide en arabe, rficfipa en grec . . . d’oii chiffre (qui derive
plutOt, suivant nous, de l’Hbbreu Seplier , compter.”)

6 “Compilatus a Magistro Jacobo de Florentia apud montem pesal
lanum,” and described by G. Land in his Catalogus codicum manu-r
scriptorum qui in bibliotheca Riccardiana Florentioc adservantur. See
Fazzari, loc. cit., p. 5.

THE IIINdU-ARABIC NUMERALS

(1307) it is called zeuero,1 while in an arithmetic of Gio-
vanni di Danti of Arezzo (1370) the word appears as
geuero2 Another form is zepiro ,3 which was also a step
from zephirum to zero.4

Of course the English cipher, French chiffre , is derived
from the same Arabic word, as-sifr , but in several lan-
guages it has come to mean the numeral figures in general.
A trace of this appears in our word ciphering , meaning
figuring or computing.5 Johann Huswirt6 uses the word
with both meanings ; he gives for the tenth character
the four names theca, cir cuius , cifra, and figura nihili.
In this statement Huswirt probably follows, as did many
writers of that period, the Algorismus of Johannes de
Sacrobosco (c. 1250 a.d.), who was also known as John
of Halifax or John of Holy wood. The commentary of

1 “Et doveto sapere cliel zeuero per se solo non significa nulla ma
6 potentia di fare significare, . . . Et decina o centinaia o migliaia
non si puote scrivere senza questo segno 0. la quale si cliiama zeuero.”
[Eazzari, loc. cit., p. 5.]

2 Ibid., p. 6.

3 Avicenna (980-1036), translation by Gasbarri et Francois, “piu il
punto (gli Arabi adoperavano il punto in vece dello zero il cui segno
0 in arabo si cliiama zepiro donde il vocabolo zero), che per sb stesso
non esprime nessun numero.” This quotation is taken from D.C.
Martines, Origine e progressi dell' aritmetica, Messina, 1865.

4 Leo Jordan, “Materialien zur Gescliichte der arabiscken Zalil-
zeichen in Frankreich,” Archiv fur Kulturgeschichte, Berlin, 1905,

pp. 155-195, gives the following two schemes of derivation, (1) “zefiro,
zeviro, zeiro, zero,” (2) “zefiro, zefro, zevro, zero.”

6 Ivobel (1518 ed., f. A4) speaks of the numerals in general as “die
der gemain man Zyfer nendt.” Kecorde ( Grounde of Artes, 1558 ed.,
f. B6) says that the zero is “called priuatly a Cyphar, though all the
other sometimes be likewise named.”

6 “Decimo X 0 theca, circul? cifra sive figura nihili appelat'.”
[Enchiridion Algorismi , Cologne, 1501.] Later, “quoniam de integris
tain in cifris quam in proiectilibus,” — the word proiectilibus referring
to markers “thrown” and used on an abacus, whence the French
jetons and the English expression “to cast an account.”

THE SYMBOL ZERO

Petrus de Dacia1 (c. 1291 a.d.) on the Algorismus vul-
garis of Sacrobosco was also widely used. The wide-
spread use of this Englishman’s work on arithmetic in
the universities of that time is attested by the large num-
ber 2 of MSS. from the thirteenth to the seventeenth cen-
tury still extant, twenty in Munich, twelve in Vienna,
thirteen in Erfurt, several in England given by Halli-
well,3 ten listed in Coxe’s Catalogue of the Oxford College
Library , one in the Plimpton collection,4 one in the
Columbia University Library, and, of course, many
others.

From as-sifr has come zephyr, cipher , and finally the
abridged form zero. The earliest printed work in which
is found this final form appears to be Calandri’s arith-
metic of 1491, 5 while in manuscript it appears at least as
early as the middle of the fourteenth century.6 It also
appears in a work, Le Kadran des mar chans, by Jelian

1 “ Decima vero o clicitur teca, circulus, vel cyfra vel figura nichili.”
[Maximilian Curtze, Petri Philomeni de Dacia in Algorismum Vulga-
rem Johannis de Sacrobosco commentarius , una cum Algorismo ipso,
Copenhagen, 1897, p. 2.] Curtze cites five manuscripts (fourteenth
and fifteenth centuries) of Dacia’s commentary in the libraries at
Erfurt, Leipzig, and Salzburg, in addition to those given by Enestrom,
Ofversigt af Kongl. Vetenskaps-Alcademiens For handling ar, 1885, pp.
15-27, 65-70; 1886, pp. 57-60.

2 Curtze, loc. cit., p. vi.

3 Rara Mathematica , London, 1841, chap, i, “Joannis de Sacro-
Bosco Tractatus de Arte Numerandi.”

4 Smith, Rara Arithmetica , Boston, 1909.

6 In the 1484 edition, Borghi uses the form “ §efiro : ouero nulla : ”
while in the 1488 edition he uses “zefiro: ouero nulla,” and in the
1540 edition, f. 3, appears “Cliiamata zero, ouero nulla.” Woepcke
asserted that it first appeared in Calandri (1491) in this sentence :
“Sono dieci le figure con le quali ciascuno numero si pud significare :
delle quali n’ d una che si chiama zero : et per se sola nulla significa”
(f. 4). [See Propagation, p. 522.]

6 Boncompagni Bulletino , Yol. XYI, pp. 673-085.

TIIE IIINDU-ARABIC NUMERALS

Certain,1 written in 1485. This word soon became fairly
well known in Spain 2 and France.3 The medieval writers
also spoke of it as the sipos ,4 and occasionally as the
wheel? cir cuius 6 (in German das Ringlein 7), circular

1 Leo Jordan, loc. cit. In the Catalogue of MSS., Bibl. de V Arsenal,
Vol. Ill, pp. 154-155, this work is No. 2904 (184 S. A. F.), Bibl. Nat.,
and is also called Petit traietA de algorisme.

2 Texada (1546) says that there are “ nueue letros yvn zero o cifra ”
(f.3).

8 Savonne (1563, 1751 ed., f. 1): “Vne ansi formee (o) qui s’appelle
nulle, & entre marchans zero,” showing the influence of Italian names
on French mercantile customs. Trenchant (Lyons, 1566, 1578 ed., p.
12) also says : “ La derniere qui s’apele nulle, ou zero ; ” but Champe-
nois, his contemporary, writing in Paris in 1577 (although the work
was not published until 1578), uses “cipher,” the Italian influence
showing itself less in this center of university culture than in the com-
mercial atmosphere of Lyons.

4 Thus Radulph of Laon (c. 1100): “Inscribitur in ultimo ordine et
figura O sipos nomine, quae, licet numerum nullum signitet, tan-

tum ad alia quaedam utilis, ut insequentibus declarabitur.” [“Der
Arithmetische Tractat des Radulph von Laon,” Abhandlungen zur Ge-
schichte der Mathematik , Vol. V, p. 97, from a manuscript of the thir-
teenth century.] Cliasles ( Comptes rendus, t. 16, 1843, pp. 1393, 1408)
calls attention to the fact that Radulph did not know how to use the
zero, and he doubts if the sipos was really identical with it. Radulph

says: “. . . figuram, cui sipos nomen est o. in motion rotulae for-

matam nullius numeri significatione inscribi solere praediximus,” and
thereafter uses rotula. He uses the sipos simply as a kind of marker
on the abacus.

6 Rabbi ben Ezra (1092-1168) used both ^3, galgal (the Hebrew for
wheel), and N1DD, sifra. See M. Steinschneider, “ Die Mathematik bei
den Juden,” in Bibliotheca Mathematica, 1893, p. 69, and Silberberg,
Das Bueh der Zahl des II. Abraham ibn Esra, Frankfurt a. M., 1895, p.
96, note 23 ; in this work the Hebrew letters are used for numerals with
place value, having the zero.

G E.g., in the twelfth-century Liber algorismi (see Boncompagni’s
Trattati, II, p. 28). So Ramus ( Libri II, 1569 ed., p. 1) says: “Cir-
culus quie nota est ultima: nil per se significat.” (See also the Sclio-
nerus ed. of Ramus, 1586, p. 1.)

7 “Und wirt das ringlein o. die Ziffer genant die niclits bedeut.”
[Kobel’s Bechenbuch, 1549 ed., f. 10, and other editions.]

THE SYMBOL ZERO

note} theca} long supposed to be from its resemblance to
the Greek theta, but explained by Petrus de Dacia as being
derived from the name of the iron 3 used to brand thieves
and robbers with a circular mark placed on the forehead or
on the cheek. It was also called omicron 4 (the Greek o),
being sometimes written b or c p to distinguish it from the
letter o. It also went by the name null 5 (in the Latin books

1 1.e. “circular figure,” our word notation having come from the
medieval nota. Thus Tzwivel (1507, f. 2) says: “Nota autem circula-
ris .o. per se sumpta nihil vsus habet. alijs tamen adiuncta earum
significantiam et auget et ordinem permutat quantum quo ponit ordi-
nem. vt adiuncta note binarij hoc modo 20 facit earn significare bis
decern etc.” Also (ibid., f. 4), “figura circularis,” “ circularis nota.”
Clichtoveus (1503 ed., f. xxxvii) calls it “nota aut circularis o,”
“ circularis nota,” and “figura circularis.” Tonstall (1522, f. B3) says
of it: “Decimo uero nota ad formam .O- litterm circulari figui'a est:
quam alij circulum, uulgus cyphram uocat,” and later (f. C4) speaks
of the “circulos.” Grammateus, in his Algorismus de integris (Erfurt,
1523, f . A2), speaking of the nine significant figures, remarks : “ His au-
tem superadditur deciina figura circularis ut 0 existens que ratione sua
nihil significat.” Noviomagus (De Numeris libri II, Paris, 1539, chap,
xvi, “De notis numerorum, quas zyphras vocant”) calls it “circularis
nota, quam ex his solam, alij sipheram, Georgius Yalla zypliram.”

2 Huswirt, as above. Ramus ( Scholae mathematicae, 1509 ed., p. 112)
discusses the name interestingly, saying: “Circulum appellamus cum
multis, quam alii thecam, alii figuram niliili, alii figuram privationis,
seu figuram nullam vocant, alii ciphram, cum tamen hodie omnes lue
node vulgo ciphrse nominentur, & his notis niunerare idem sit quod
ciphrare.” Tartaglia (1592 ed., f. 9) says: “si chiama da alcuni tecca,
da alcuni circolo, da altri cifra, da altri zero, & da alcuni altri nulla.”

3 “Quare autem aliis nominibus vocetur, non dicit auctor, quia
omnia alia nomina habent rationem suae lineationis sive figuration is.
Quia rotunda est, dicitur haec figura teca ad similitudinem tecae.
Teca enim est ferrum figurae rotundae, quod ignitum solet in quibus-
dam regionibus imprimi fronti vel maxillae furis seu latronum.” [Loc.
cit., p. 26.] But in Greek theca (OHKH, 017*77) is a place to put some-
thing, a receptacle. If a vacant column, e.g. in the abacus, was so
called, the initial might have given the early forms © and 0 for the zero.

4 Buteo, Logistica, Lyons, 1559. See also Wertheim in the Biblio-
theca Mathematica, 1901, p. 214.

5 “ O est appellee chiffre ou nulle ou figure de nulle valeur.” [La
Roche, L' arithimZtique, Lyons, 1520.]

THE HINDU-ARABIC NUMERALS

nihil 1 or nulla? and in the French rim 3), and very com-
monly by the name cipher .4 Wallis 5 gives one of the earli-
est extended discussions of the various forms of the word,
giving certain other variations worthy of note, as ziphra , zi-
fera , siphra , ciphra , tsiphra, tziphra , and the Greek r&'cfypa.6

1 “ Decima autem figura nihil uocata,” “ figura nihili (quam etiarn
cifram uocant).” [Stifel, Arithmetica Integra , 1544, f. 1.]

2 “ Zifra, & Nulla uel ligura Nihili.” [Scheubel, 1545, p. 1 of cli. 1.]
Nulla is also used by Italian writers. Thus Sfortunati (1545 ed., f. 4)
says : “ et la decima nulla & e cliiamata questa decima zero ; ” Cataldi
(1602, p. 1): “La prima, che b o, si chiama nulla, ouero zero, ouero
niente.” It also found its way into the Dutch arithmetics, e.g. Raets
(1576, 1580 ed., f. A3): “Nullo dat ist niet;” Van der Schuere (1600,
1624 ed., f. 7); Wilkens (1669 ed., p. 1). In Germany Johann Albert
(Wittenberg, 1534) and Rudolff (1526) both adopted the Italian nulla
and popularized it. (See also Ivuckuck, Die Eechenkunst im sechzehn-
ten Jahrhundert , Berlin, 1874, p. 7 ; Gunther, Geschichte, p. 316.)

8 “La dixifeme s’appelle chifre vulgairement : les vns l’appellant
zero : nous la pourrons appeller vn Rien.” [Peletier, 1607 ed., p. 14.]

4 It appears in the Polish arithmetic of Rios (1538) as cyfra. “The
Ciphra 0 augmenteth places, but of liimselfe signifieth not,” Digges,
1579, p. 1. Hodder (10th ed., 1672, p. 2) uses only this word (cypher
or cipher), and the same is true of the first native American arithme-
tic, written by Isaac Greenwood (1729, p. 1). Petrus de Dacia derives
cyfra from circumference. “Vocatur etiam cyfra, quasi circumfacta
vel circumferenda, quod idem est, quod circulus non liabito respectu
ad centrum.” [Loc. cit., p. 26.]

5 Opera mathematical 1695, Oxford, Yol. I, chap, ix, Mathesis univer-
salis, “De figuris numeralibus,” pp. 46-49 ; Yol. II, Algebra , p. 10.

c Martin, Origine de notre systeme de numeration icrite , note 149, p. 36
of reprint, spells rulcppa from Maximus Planudes, citing Wallis as an
authority. This is an error, for Wallis gives the correct form as above.

Alexander von Humboldt, “fiber die bei verschiedenen Volkern
iiblichen Systeme von Zalilzeicben und fiber den Ursprung des Stellen-
werthes in den indischen Zalilen,” Crelle’s Journal fur reine und
angewandte Mathematik, Yol. IV, 1829, called attention to the work
apidpol’lvSiKOL of the monk Neophytos, supposed to be of the four-
teenth century. In this work the forms rptuppa and rtfpfipa appear.
See also Boeckh, De abaco Graecorum, Berlin, 1841, and Tannery, “Le
Scliolie du moine Neophytos,” lleme ArcMologique , 1885, pp. 99-102.
Jordan, loc. cit., gives from twelfth and thirteenth century manuscripts
the forms cifra , ciffre, chifras, and cifrus. Du Cange, Glossarium mediae
et ivfimae Latinitatis , Paris, 1842, gives also chilerae. Dasypodius,
Institutiones Mathematicae, Strassburg, 1593-1596, adds the forms
zyphra and syphra. Boissifere, Dart d'arylhmctique contenant toute
dimention , tres-singulier et commode , tant pour l' art militaire que autrcs
calculations , Paris, 1554: “Puis y en a vn autre diet zero lequel ne
designe nulle quantity par soy, ains seulement les loges vuides,”

CHAPTER V

the question of the introduction of the

NUMERALS INTO EUROPE BY BOETHIUS

Just as we were quite uncertain as to the origin of
the numeral forms, so too are we uncertain as to the
time and place of their introduction into Europe. There
are two general theories as to this introduction. The first
is that they were carried by the Moors to Spain in the
eighth or ninth century, and thence were transmitted
to Christian Europe, a theory which will be considered
later. The second, advanced by Woepcke,1 is that they
were not brought to Spain by the Moors, but that they
were already in Spain when the Arabs arrived there, having
reached the West through the Neo-Pythagoreans. There
are two facts to support this second theory : (1) the forms
of these numerals are characteristic, differing materially
from those which were brought by Leonardo of Pisa
from Northern Africa early in the thirteenth century
(before 1202 a.d.) ; (2) they are essentially those which

1 Propagation , pp. 27, 234, 442. Treutlein, “Das Rechnen im 10.
Jalirliundert,” Abhandlungen zur Geschichte der Mathematik, Yol. I,
p. 5, favors the same view. It is combated by many writers, e.g. A.C.
Burnell, loc. cit., p. 59. Long before Woepcke, I. F. and G.I.Weid-
ler, Be characteribus numerorum vulgaribns et eorum aetatibus , Witten-
berg, 1727, asserted the possibility of their introduction into Greece
by Pythagoras or one of his followers : “ Potuerunt autem ex oriente,
uel ex Phoenicia, ad graecos traduci, uel Pythagorae, uel eius discipu-
lorum auxilio, cum aliquis co, proficiendi in literis causa, iter faceret,
et hoc quoque inuentum addisceret.”

TIIE HINDU-ARABIC NUMERALS

tradition has so persistently assigned to Boethius (c. 500
A.D.), and which lie would naturally have received, if
at all, from these same Neo-Pythagoreans or from the
sources from which they derived them. Furthermore,
W oepcke points out that the Arabs on entering Spain
(711 A.d.) would naturally have followed their custom
of adopting for the computation of taxes the numerical
systems of the countries they conquered,1 so that the
numerals brought from Spain to Italy, not having under-
gone the same modifications as those of the Eastern Arab
empire, would have differed, as they certainly did, from
those that came through Bagdad. The theory is that the
Hindu system, without the zero, early reached Alexan-
dria (say 450 a.d.), and that the Neo-Pytliagorean love
for the mysterious and especially for the Oriental led
to its use as something bizarre and cabalistic ; that it
was then passed along the Mediterranean, reaching Boe-
thius in Athens or in Rome, and to the schools of Spain,
being discovered in Africa and Spain by the Arabs even
before they themselves knew the improved system with
the place value.

1 E.g., they adopted the Greek numerals in use in Damascus and
Syria, and the Coptic in Egypt. Tlieophanes (758-818 a.d.), Chrono-
graphia, Scriptores Ilistoriae Byzantinae, Vol. XXXIX, Bonnae, 1839,
p. 575, relates that in 699 a.d. the caliph Walld forbade the use of the
Greek language in the bookkeeping of the treasury of the caliphate,
but permitted the use of the Greek alphabetic numerals, since the
Arabs had no convenient number notation : Kal iKti\v<re ypd<pe<rOai 'EX-
'KyvLcrrl robs bypoalovs tOjv 'koyoOealwv kwSlkcls , aXX’ ’Apafilois aura irapaay-
palvetrOcu, xwP'LS T&v 'Pv4>wvi eireLdy aSi ivarov ry eKelvwv yXi&crcry povaSa y
SvdSa y rpiada y 6ktcj ypuxv y rpla ypdcpeadai • Stb Kal £ios cryp.epbv elcriv
<rbv ai/Tois vordpioi XpurTLavot. The importance of this contemporaneous
document was pointed out by Martin, loc. cit. Karabacek, “Die In-
volutio im arabischen Schriftwesen,” Vol.CXXXV of Sitzungsberichte
cl. phil.-ldst. Classe d. k. Alcad. d. Wiss., Vienna, 1896, p. 25, gives an
Arabic date of 868 a.d. in Greek letters.

TIIE BOETIIIUS QUESTION

A recent theory set forth by Bubnov 1 also deserves
mention, chiefly because of the seriousness of purpose
shown by this well-known writer. Bubnov holds that
the forms first found in Europe are derived from ancient
symbols used on the abacus, but that the zero is of Hindu
origin. This theory does not seem tenable, however, in
the light of the evidence already set forth.

Two questions are presented by Woepcke’s theory:
(1) What was the nature of these Spanish numerals, and
how were they made known to Italy ? (2) Did Boethius
know them ?

The Spanish forms of the numerals were called the
huruf al-c/obar, the gobar or dust numerals, as distin-
guished from the huruf al-jumal or alphabetic numer-
als. Probably the latter, under the influence of the
Syrians or Jews,2 were also used by the Arabs. The
significance of the term gobar is doubtless that these
numerals were written on the dust abacus, this plan
being distinct from the counter method of representing
numbers. It is also worthy of note that Al-BirunI 'states
that the Hindus often performed numerical computations
in the sand. The term is found as early as c. 950,
in the verses of an anonymous writer of Kairwan, in
Tunis, in which the author speaks of one of his works
on gobar calculation ; 3 and, much later, the Arab writer
Abu Bekr Mohammed ibn 'Abdallah, surnamed al-Hassar

1 The Origin and History of Our Numerals (in Russian), Kiev, 1908 ;
The Independence of European Arithmetic (in Russian), Kiev.

2 Woepcke, loc. cit., pp. 462, 262.

3 Woepcke, loc. cit., p. 240. Hisab-al-Gobdr , by an anonymous
author, probably Abu Sabi Dunash ibn Tamim, is given by Stein-
schneider, “ Die Mathematik bei den Juden,” Bibliotheca Mathematical
1895, p. 26,

THE IiINDU-ARABIC NUMERALS

(the arithmetician), wrote a work of which the second
chapter was “ On the dust figures.” 1

The gohar numerals themselves were first made known
to modern scholars by Silvestre de Sacy, who discovered
them in an Arabic manuscript from the library of the
ancient abbey of St.-Germain-des-Pres.2 The system has
nine characters, but no zero. A dot above a character
indicates tens, two dots hundreds, and so on, 5 meaning
50, and 5 meaning 5000. It has been suggested that
possibly these dots, sprinkled like dust above the numer-
als, gave rise to the word gobdr,2, but this is not at all
probable. This system of dots is found in Persia at a
much later date with numerals quite like the modern
Arabic ; 4 but that it was used at all is significant, for it
is hardly likely that the western system would go back to
Persia, when the perfected Hindu one was near at hand.

At first sight there would seem to be some reason for
believing that this feature of the gobar system was of

1 Steinschneider in the Abhandlungen, Yol. Ill, p. 110.

2 See his Grammaire arabe , Yol. I, Paris, 1810, plate VIII ; Ger-
liardt, Etudes, pp. 9-11, and Entstehung etc., p. 8; I. F. Weidler,
Spicilegium observationum ad historiam notarum numeralium perti-
nentium , Wittenberg, 1755, speaks of the “figura cifrarum Saracenica-
rum ” as being different from that of the “ characterum Boethianorum,”
which are similar to the “ vulgar ’ ’ or common numerals ; see also Hum-
boldt, loc. cit.

8 Gerhardt mentions it in his Entstehung etc., p. 8 ; Woepcke, Pro-
pagation, states that these numerals were used not for calculation, but
very much as we use Roman numerals. These superposed dots are
found with both forms of numerals ( Propagation , pp. 244-246).

4 Gerhardt ( Etudes , p. 9) from a manuscript in the Bibliothfeque

Nationale. The numeral forms are Q A V

g A V l| O S

indicated by JJ and 200 by j). This scheme of zero dots was also

adopted by the Byzantine Greeks, for a manuscript of Planudes in the
Bibliotk&que Nationale has numbers like ka for 8,100,000,000. See
Gerhardt, Etudes, p. 19. Pihan, Expose etc., p. 208, gives twp forms,
Asiatic and Maghrebian, of “Ghobar” numerals.

THE BOETHIUS QUESTION

Arabic origin, and that the present zero of these people,1
the dot, was derived from it. It was entirely natural that
the Semitic people generally should have adopted such a
scheme, since their diacritical marks would suggest it,
not to speak of the possible influence of the Greek
accents in the Hellenic number system. When we con-
sider, however, that the dot is found for zero in the
Bakhsall manuscript,2 and that it was used in subscript
form in the Kitab al-Fihrist 3 in the tenth century, and as
late as the sixteenth century,4 although in this case prob-
ably under Arabic influence, we are forced to believe that
this form may also have been of Hindu origin.

The fact seems to be that, as already stated,6 the Arabs
did not immediately adopt the Hindu zero, because it
resembled their 5 ; they used the superscript dot as
serving their purposes fairly well ; they may, indeed,
have carried this to the west and have added it to the
gobar forms already there, just as they transmitted it
to the Persians. Furthermore, the Arab and Hebrew
scholars of Northern Africa in the tenth century knew
these numerals as Indian forms, for a commentary on
the Sefer Yeslrdh by Abu Sahl ibn Tamim (probably
composed at Kairwan, c. 950) speaks of “the Indian
arithmetic known under the name of gobar or dust cal-
culation.” 6 All this suggests that the Arabs may very

1 See Chap. IV.

2 Possibly as early as the third century a.d., but probably of the
eighth or ninth. See Cantor, I (3), p. 598.

3 Ascribed by the Arabic writer to India.

4 See Woepcke’s description of a manuscript in the Chasles library,
“Recherches sur l’histoire des sciences math&hatiques chez lesorien-
taux,” Journal Asiatique, IV (5), 1859, p. 358, note.

5 P. 56.

6 Reinaud, Mtmoire sur Vlnde, p. 399. In the fourteenth century
one Siliab al-DIn wrote a work on which a scholiast to the Bodleian

THE IIINDU-ARABIC NUMERALS

likely have known the gobar forms before the numerals
reached them again in 773.1 The term “gobar numer-
als ” was also used without any reference to the peculiar
use of dots.2 In this connection it is worthy of mention
that the Algerians employed two different forms of
numerals in manuscripts even of the fourteenth cen-
tury,3 and that the Moroccans of to-day employ the
European forms instead of the present Arabic.

The Indian use of subscript dots to indicate the tens,
hundreds, thousands, etc., is established by a passage hi
the Kitab al-Fihrist 4 (987 a.d.) in which the writer dis-
cusses the written language of the people of India. Not-
withstanding the importance of this reference for the
early history of the numerals, it has not been mentioned
by previous writers on this subject. The numeral forms
given are those which have usually been called Indian,6
in opposition to gobar. In this document the dots are
placed below the characters, instead of being superposed
as described above. The significance was the same.

In form these gobar numerals resemble our own much
more closely than the Arab numerals do. They varied
more or less, but were substantially as follows :

manuscript remarks : “ Tlie science is called Algobar because the
inventor had the habit of writing the figures on a tablet covered with
sand.” [Gerhardt, Ktudes, p. 11, note.]

1 Gerhardt, Entstehung etc., p. 20.

2 H. Suter, “Das Rechenbuch des Abu Zakarija el-Hassar,” Bibli-
otheca Mathematical Vol. II (3), p. 15.

8 A. Devoulx, “ Les chiffres arabes,” Revue Africaine, Vol. XVI, pp.
455-458.

4 Kitab al-Fihrist , G. Fliigel, Leipzig, Vol. I, 1871, and Vol. II,
1872. This work was published after Professor Fliigel’s death by J.
Roediger and A. Mueller. The first volume contains the Arabic text
and the second volume contains critical notes upon it.

8 Like those of line 5 in the illustration on page G9,

THE BOETHIUS QUESTION

i J g 1 J ^ 1

■ jlHUiW

‘O ■/ V } i / ^ M

4 ' y 9 ;? £ $- ^>Z I

s ; /» v 'i <3 H J

6 6 ? f * l 7

The question of the possible influence of the Egyptian
demotic and hieratic ordinal forms has been so often
suggested that it seems well to introduce them at this
point, for comparison with the gobar forms. They would
as appropriately be used in connection with the Hindu
forms, and the evidence of a relation of the first three
with all these systems is apparent. The only further
resemblance is in the Demotic 4 and in the 9, so that the
statement that the Hindu forms in general came from

1 Woepcke, Recherches sur Vhistoire des sciences matMmatiques chez
les orientciux, loc. cit. ; Propagation , p. 57.

2 Al-Hassar’s forms, Suter, Bibliotheca Mathematical Yol. II (3),
p. 15.

3 Woepcke, Sur une donnie historique , etc., loc. cit. The name gobar
is not used in the text. The manuscript from which these are taken
is the oldest (970 a.d.) Arabic document known to contain all of the
numerals.

4 Silvestre de Sacy, loc. cit. He gives the ordinary modern Arabic
forms, calling them Indian.

5 and 6 Woepcke, “Introduction au calcul Gobari et Ilawal,” Atti
dell ’ accademia pontificia dei nuovi Lincei, Yol. XIX. The adjective ap-
plied to the forms in 6 is gobari and to those in 6 indienne. This is the
direct opposite of Woepcke’s use of these adjectives in the Recherches
sur Vhistoire cited above, in which the ordinary Arabic forms (like
those in row 1 2 3 4 5) are called indiens.

These forms are usually written from right to left.

70 THE HINDU- ARABIC NUMERALS

this source has no foundation. The first four Egyptian
cardinal numerals 1 resemble more the modern Arabic.

This theory of the very early
introduction of the numerals
into Europe fails in several
points. In the first place the
early Western forms are not
known ; in the second place *
some early Eastern forms are
like the gobar, as is seen in the
third line on p. 69, where the
forms are from a manuscript
written at Shiraz about 970 A. n.,
and in which some western Ara-
bic forms, e.g. p for 2, are also
used. Probably most significant
of all is the fact that the gobar
numerals as given by Sacy are
all, with the exception of the symbol for eight, either sin-
gle Arabic letters or combinations of letters. So much for
the W oepcke theory and the meaning of the gobar numer-
als. We now have to consider the question as to whether
Boethius knew these gobar forms, or forms akin to them.

This large question 2 suggests several minor ones :
(1) Who was Boethius? (2) Could he have known
these numerals? (3) Is there any positive or strong cir-
cumstantial evidence that he did know them ? (4) What
are the probabilities in the case ?

1 J. G. Wilkinson, The Manners and Customs of the Ancient Egyp-
tians, revised by S. Birch, London, 1878, Vol. II, p. 493, plate XVI.

2 There is an extensive literature on this “ Boethius-Frage.” The
reader who cares to go fully into it should consult the various volumes
of the Jahrbucli iiber die Fortschritte der Mathematik.

1 A
21

11, 44

>.y

II
11

Demotic and Hieratic
Ordinals

THE BOETHIUS QUESTION

First, who was Boethius, — Divus1 Boethius as he was
called in the Middle Ages ? Anicius Manlius Severinus
Boethius2 was born at Rome c. 475. He was a mem-
ber of the distinguished family of the Anicii,3 which had
for some time before his birth been Christian. Early
left an orphan, the tradition is that he was taken to
Athens at about the age of ten, and that he remained
there eighteen years.4 He married Rusticiana, daughter
of the senator Symmachus, and this union of two such
powerful families allowed him to move in the highest
circles.6 Standing strictly for the right, and against all
iniquity at court, he became the object of hatred on the
part of all the unscrupulous element near the throne,
and his bold defense of the ex-consul Albinus, unjustly
accused of treason, led to his imprisonment at Pavia6
and his execution in 524.7 Not many generations after
his death, the period being one in which historical criti-
cism was at its lowest ebb, the church found it profitable
to look upon his execution as a martyrdom.8 He was

1 This title was first applied to Roman emperors in posthumous
coins of Julius Ctesar. Subsequently the emperors assumed it during
their own lifetimes, thus deifying themselves. See F. Gnecchi, Monete
romane, 2d ed., Milan, 1900, p. 299.

2 This is the common spelling of the name, although the more cor-
rect Latin form is Boetius. See Harper’s Diet, of Class. Lit. and
Antiq., New York, 1897, Vol. I, p. 213. There is much uncertainty as
to his life. A good summary of the evidence is given in the last two
editions of the Encyclopedia Britannica.

8 Ilis father, Flavius Manlius Boethius, was consul in 487.

4 There is, however, no good historic evidence of this sojourn in
Athens.

6 His arithmetic is dedicated to Symmachus: “Domino suo patri-
cio Symmaclio Boetius.” [Friedlein ed., p. 3.]

6 It was while here that he wrote De consolatione philosophiae.

7 It is sometimes given as 525.

8 There was a medieval tradition that he was executed because of a
work on the Trinity.

THE IIINDU-ARABIC NUMERALS

accordingly looked upon as a saint,1 liis bones were en-
shrined,2 and as a natural consequence his books were
among the classics in the church schools for a thousand
years.3 It is pathetic, however, to think of the medieval
student trying to extract mental nourishment from a
work so abstract, so meaningless, so unnecessarily com-
plicated, as the arithmetic of Boethius.

He was looked upon by his contemporaries and imme-
diate successors as a master, for Cassiodorus4 * (c. 490-
c. 585 a.d.) says to him : “ Through your translations
the music of Pythagoras and the astronomy of Ptolemy
are read by those of Italy, and the arithmetic of Nicoma-
chus and the geometry of Euclid are known to those of ■
the West.”6 Founder of the medieval scholasticism,

1 Hence the Divas in his name.

2 Thus Dante, speaking of his burial place in the monastery of St.
Pietro in Ciel d’ Oro, at Pavia, says :

“The saintly soul, that shows
The world’s deceitfulness, to all who hear him,

Is, with the sight of all the good that is,

Blest there. The limbs, whence it was driven, lie
Down in Cieldauro ; and from martyrdom
And exile came it here.” — Paradiso, Canto X.

3 Not, however, in the mercantile schools. The arithmetic of Boe-
thius would have been about the last book to be thought of in such
institutions. While referred to by Bseda (672-735) and Hrabanus
Maurus (c. 776-856), it was only after Gerbert’s time that the Boetii
de institutione arithmetica libri duo was really a common work.

4 Also spelled Cassiodorius.

6 As a matter of fact, Boethius could not have translated any work

by Pythagoras on music, because there was no such work, but he did
make the theories of the Pythagoreans known. Neither did he trans-
late Nicomachus, although he embodied many of the ideas of the Greek
writer in his own arithmetic. Gibbon follows Cassiodorus in these
statements in his Decline and Fall of the Roman Empire , chap, xxxix.
Martin pointed out with positiveness the similarity of the first book
of Boethius to the first five books of Nicomachus. [ies signes nume-
raux etc., reprint, p. 4.]

TIIE BOETHIUS QUESTION

distinguishing; the triyium and quadrivium,1 writing the
only classics of liis time, Gibbon well called him “ the last
of the Romans whom Cato or Tully could have acknowl-
edged for their countryman.” 2

The second question relating to Boethius is this : Could
he possibly have known the Hindu numerals ? In view
of the relations that will be shown to have existed be-
tween the East and the West, there can only be an
affirmative answer to this question. The numerals had
existed, without the zero, for several centuries ; they
had been well known in India ; there had been a contin-
ued interchange of thought between the East and W est ;
and warriors, ambassadors, scholars, and the restless trader,
all had gone back and forth, by land or more frequently
by sea, between the Mediterranean lands and the centers
of Indian commerce and culture. Boethius could very
well have learned one or more forms of Hindu numerals
from some traveler or merchant.

To justify this statement it is necessary to speak more
fully of these relations between the Far East and Europe.
It is true that we have no records of the interchange of
learning, in any large way, between eastern Asia and
central Europe in the century preceding the time of
Boethius. But it is one of the mistakes of scholars to
believe that they are the sole transmitters of knowledge.

1 The general idea goes back to Pythagoras, however.

2 J. C. Scaliger in his Po'etice also said of him: “Boethii Severini
ingenium, eruditio, ars, sapientia facile provocat omnes auctores, sive
illi Graeci sint, sive Latini” [Heilbronner, Hist. math, univ., p. 387].
Libri, speaking of the time of Boethius, remarks: “Nous voyons du
temps de Th^odoric, les lettres repi’endre une nouvelle vie en Italie, les
^colesflorissantesetlessavans honoris. Etcerteslesouvragesde Boece,
de Cassiodore, de Symmaque, surpassent de beaucoup toutes les produc-
tions du si tele prbccident.” [ Ilistoire des mathdmatiques , Vol. I, p. 78.]

THE HINDU- A II ABIC NUMERALS

As a matter of fact there is abundant reason for believ-
ing that Hindu numerals would naturally have been
known to the Arabs, and even along every trade route
to the remote west, long before the zero entered to make
their place-value possible, and that the characters, the
methods of calculating, the improvements that took place
from time to time, the zero when it appeared, and the
customs as to solving business problems, would all have ,
been made known from generation to generation along
these same trade routes from the Orient to the Occident.
It must always be kept in mind that it was to the trades-
man and the wandering scholar that the spread of such
learning was due, rather than to the school man. Indeed,
Avicenna1 (980-1037 a.d.) in a short biography of him-
self relates that when his people were living at Bokhara
his father sent him to the house of a grocer to learn the
Hindu art of reckoning, in which this grocer (oil dealer,
possibly) was expert. Leonardo of Pisa, too, had a similar
training.

The whole question of this spread of mercantile knowl-
edge along the trade routes is so connected with the go-
bar numerals, the Boethius question, Herbert, Leonardo
of Pisa, and other names and events, that a digression
for its consideration now becomes necessary.2

1 Carra de Vaux, Avicmne , Paris, 1900; Woepcke, Sur Vintroduc-
tion, etc.; Gerhard t, Entsteliung etc., p. 20. Avicenna is a corruption
from Ibn Slna, as pointed out by Wiistenfeld, Geschichte dcr arabischen
Aerzte und Naturforschcr, Gottingen, 1840. Ilis full name is Abu All
al-Hosein ibn Sink. For notes on Avicenna’s arithmetic, see Woepcke,
Propagation , p. 502.

2 On the early travel between the East and the West the follow-
ing works may be consulted: A. Hillebrandt, Alt-Indien , containing
“Cliinesische Reisende in Indien,” Breslan, 1899, p. 179 ; C. A. Skeel,
Travel in the First Century after Christ , Cambridge, 1901, p. 142; M.
Rcinaud, “ Relations politiques ct commerciales de l’empire romain

THE BOETHIUS QUESTION

Even in very remote times, before the Hindu numer-
als were sculptured in the cave of Nana Ghat, there were
trade relations between Arabia and India. Indeed, long
before the Aryans went to India the great Turanian race
had spread its civilization from the Mediterranean to the
Indus.1 At a much later period the Arabs were the inter-
mediaries between Egypt and Syria on the west, and the
farther Orient.2 In the sixth century b.c., 1 lecatams,3
the father of geography, was acquainted not only with the
Mediterranean lands but with the countries as far as the
Indus,4 and in Biblical times there were regular triennial
voyages to India. Indeed, the story of Joseph bears
witness to the caravan trade from India, across Arabia,
and on to the banks of the Nile. About the same time
as Iiecatteus, Scylax, a Persian admiral under Darius,
from Caryanda on the coast of Asia Minor, traveled to

avec l’Asie orientale,” in the Journal Asiatique , Mars-Avril, 1863,
Vol. I (6), p. 93; Beazley, Dawn of Modern Geography , a History of
Exploration and Geographical Science from the Conversion of the Roman
Empire to A.D. 1420 , London, 1897-1906, 3 vols. ; Ileyd, Geschichte des
Levanthandels im Mittelalter , Stuttgart, 1897 ; J. Keane, The Evolution
of Geography , London, 1899, p. 38 ; A. Cunningham, Corpus inscriptio-
num Indicai-um , Calcutta, 1877, Yol. I; A. Neander, General History
of the Christian Religion and Church , 5th American ed., Boston, 1855,
Yol. Ill, p. 89; R. C. Dutt, A History of Civilization in Ancient
India , Vol. II, Bk. V, chap, ii ; E. C. Bayley, loc. cit., p. 28 et seq.;
A. C. Burnell, loc. cit., p. 3; J. E. Tennent, Ceylon , London, 1859,
Vol. I, p. 159; Geo. Tumour, Epitome of the History of Ceylon , Lon-
don, n.d., preface; “ Philalethes,” History of Ceylon , London, 1816,
chap, i; II. C. Sirr, Ceylon and the Cingalese , London, 1850, Vol. I,
chap. ix. On the Hindu knowledge of the Nile see E. Wilford, Asi-
atick Researches, Vol. Ill, p. 295, Calcutta, 1792.

1 G. Oppert, On the Ancient Commerce of India , Madras, 1879, p. 8.

2 Gerhardt, Etudes etc., pp. 8, 11.

3 See Smith’s Dictionary of Greek and Roman Biography and Mythol-
ogy.

4 P. M. Sykes, Ten Thousand Miles in Persia , or Eight Years in
Irdn, London, 1902, p. 167. Sykes was the first European to follow
the course of Alexander’s army across eastern Persia.

THE IIINDU-ARABIC NUMERALS

northwest India and wrote upon his ventures.1 lie induced
the nations along the Indus to acknowledge the Persian
supremacy, and such number systems as there were in
these lands would naturally have been known to a man
of his attainments.

A century after Scylax, Herodotus showed consider-
able knowledge of India, speaking of its cotton and its
gold,2 telling how Sesostris3 fitted out ships to sail to
that country, and mentioning the routes to the east.
These routes were generally by the Red Sea, and had
been followed by the Phoenicians and the Sabseans, and
later were taken by the Greeks and Romans.4

In the fourth century n.c. the W est and East came into
very close relations. As early as 330, Pytheas of Mas-
silia (Marseilles) had explored as far north as the north-
ern end of the British Isles and the coasts of the German
Sea, while Macedon, in close touch with southern France,
was also sending her armies under Alexander6 through
Afghanistan as far east as the Punjab.6 Pliny tells us

that Alexander the Great employed surveyors to measure

•

1 Biihler, Indian Brahma Alphabet, note, p. 27 ; Palaeographie, p. 2 ;
Herodoti Halicarnassei historia, Amsterdam, 1763, Bk. IV, p. 300;
Isaac Vossius, Periplus Scylacis Caryandensis, 1639. It is doubtful
whether the work attributed to Scylax was written by him, but in
any case the work dates back to the fourth century b.c. See Smith’s
Dictionary of Greek and Roman Biography.

2 Herodotus, Bk. III.

8 Rameses II(?), the Sesoosis of Diodorus Siculus.

4 Indian Antiquary, Vol. I, p. 229; F. B. Jevons, Manual of Greek
Antiquities, London, 1895, p. 386. On the relations, political and com-
mercial, between India and Egypt c. 72 b.c., under Ptolemy Auletes,
see the Journal Asiatique, 1863, p. 297.

6 Sikandar, as the name still remains in northern India.

G Harper's Classical Diet., New York, 1897, Vol. I, p. 724; F. B.
Jevons, loc. cit., p. 389; J. C. Marshman, Abridgment of the History
of India, chaps, i and ii.

THE BOETHIUS QUESTION

the roads of India; and one of the great highways is
described by Megasthenes, who in 295 b.c., as the ambas-
sador of Seleucus, resided at Patallputra, the present
Patna.1

The Hindus also learned the art of coining from the
Greeks, or possibly from the Chinese, and the stores of
Greco-Hindu coins still found in northern India are a
constant source of historical information.2 The Rama-
yana speaks of merchants traveling in great caravans
and embarking by sea for foreign lands.3 Ceylon traded
with Malacca and Siam, and Java was colonized by Hindu
traders, so that mercantile knowledge was being spread
about the Indies during all the formative period of the
numerals.

Moreover the results of the early Greek invasion were
embodied by Dicsearchus of Messana (about 320 b.c.) in
a map that long remained a standard. Furthermore,
Alexander did not allow his influence on the East to
cease. He divided India into three satrapies,4 placing
Greek governors over two of them and leaving a Hindu
ruler in charge of the third, and in Bactriana, a part of
Ariana or ancient- Persia, he left governors ; and in these
the western civilization was long in evidence. Some of
the Greek and Roman metrical and astronomical terms

1 Oppert, loc. cit., p. 11. It was at or near this place that the first
great Indian mathematician, Aryabhata, was born in 476 a.d.

2 Biihler, Palaeographie , p. 2, speaks of Greek coins of a period
anterior to Alexander, found in northern India. More complete infor-
mation may be found in Indian Coins, by E. J. Rapson, Strassburg,
1898, pp. 3-7.

3 Oppert, loc. cit., p. 14 ; and to him is due other similar infor-
mation.

4 J. Beloch, Griechische Geschichte, Vol. Ill, Strassburg, 1904, pp.
30-31.

THE IIINDU-ARABIC NUMERALS

found their way, doubtless at this time, into the Sanskrit
language.1 Even as late as from the second to the fifth
centuries a.d., Indian coins showed the Hellenic influ-
ence. The Hindu astronomical terminology reveals the
same relationship to western thought, for Y araha-Mihira
(6th century A.D.), a contemporary of Aryabhata, enti-
tled a work of his the Brhat-Samhitd , a literal translation
of iie^dXrj cnWaft? of Ptolemy ; 2 and in various ways is
this interchange of ideas apparent.3 It could not have
been at all unusual for the ancient Greeks to go to In-
dia, for Strabo lays down the route, saying that all who
make the journey start from Ephesus and traverse Phrygia
and Cappadocia before taking the direct road.4 The prod-
ucts of the East were always finding their way to the
West, the Greeks getting their ginger5 from Malabar,
as the Phoenicians had long before brought gold from
Malacca.

Greece must also have had early relations with China,
for there is a notable similarity between the Greek and
Chinese life, as is shown in their houses, their domestic
customs, their marriage ceremonies, the public story-
tellers, the puppet shows which Herodotus says were
introduced from Egypt, the street jugglers, the games of
dice,6 the game of finger-guessing,7 the water clock, the

1 E.g., the denarius, the words for hour and minute (u >pa, \eirr6v),
and possibly the signs of the zodiac. [R. Caldwell, Comparative Gram-
mar of the Dravidian Languages , London, 1856, p. 438.] On the prob-
able Chinese origin of the zodiac see Schlegel, loc. cit.

2 Marie, Vol. II, p. 73 ; R. Caldwell, loc. cit.

3 A. Cunningham, loc. cit., p. 50.

4 C. A. J. Skeel, Travel , loc. cit., p. 14.

6 Inchiver, from inchi , “the green root.” [Indian Antiquary , Vol. I,
p. 352.]

6 In China dating only from the second century a.d., however.

7 The Italian morra.

TIIE BOETHIUS QUESTION

music system, the use of the myriad,1 the calendars, and

in many other ways.2 In passing through the suburbs of

Peking to-day, on the way to the Great Bell temple, one

is constantly reminded of the semi-Greek architecture of

Pompeii, so closely does modern China touch the old

classical civilization of the Mediterranean. The Chinese

historians tell us that about 200 b.c. their arms were suc-
cessful in the far west, and that in 180 b.c. an ambassador
went to Bactria, then a Greek city, and reported that Chi-
nese products were on sale in the markets there.3 There
is also a noteworthy resemblance between certain Greek
and Chinese words,4 showing that in remote times there
must have been more or less interchange of thought.

The Romans also exchanged products with the East.
Horace says, “A busy trader, you hasten to the farthest
Indies, flying from poverty over sea, over crags, over
fires.” 5 The products of the Orient, spices and jewels
from India, frankincense from Persia, and silks from
China, being more in demand than the exports from the
Mediterranean lands, the balance of trade was against
the West, and thus Roman coin found its way east-
ward. In 1898, for example, a number of Roman coins
dating from 114 b.c. to Hadrian’s time were found at
Pakli, a part of the Hazara district, sixteen miles north
of Abbottabad,6 and numerous similar discoveries have
been made from time to time.

1 J. Bowring, The Decimal System , London, 1854, p. 2.

2 II. A. Giles, lecture at Columbia University, March 12, 1902, on

“China and Ancient Greece.” 3 Giles, loc. cit.

4 E.g., the names for grape, radish ( la-po , pacpy), water-lily ( si-kua ,
“west gourds”; o-uaja, “gourds”), are much alike. [Giles, loe. cit.]

5 Epistles, I, 1, 45-46. On the Roman trade routes, 'see Beazley,
loc. cit., Vol. I, p. 179.

0 Am. Journ. of Archeol., Vol. IV, p. 366.

THE HINDU- ARABIC NUMERALS

Augustus speaks of envoys received by him from India,
a thing never before known,1 and it is not improbable that
he also received an embassy from China.2 Suetonius (first
century a.d.) speaks in his history of these relations,3 as
do several of his contemporaries,4 and Vergil5 tells of
Augustus doing battle in Persia. In Pliny’s time the
trade of the Roman Empire with Asia amounted to a
million and a quarter dollars a year, a sum far greater
relatively then than now,6 while by the time of Constan-
tine Europe was in direct communication with the Far
East.7

In view of these relations it is not beyond the range of
possibility that proof may sometime come to light to show
that the Greeks and Romans knew something of the

1 M. Perrot gives this conjectural restoration of his words: “Ad
me ex India regum legationes saepe missi sunt numquam antea visae
apud quemquam principem Romanorum.” [M. Reinaud, “Relations
politiques et commercial es de P empire romain avec l’Asie orientale,”
Journ. Asiat., Yol. I (6), p. 93.]

2 Reinaud, loc. cit., p. 189. Floras, II, 34 (IV, 12), refers to it:
“ Seres etiam habitantesque sub ipso sole Indi, cum gemmis et margari-
tis elephantes quoque inter munera trahentes nihil rnagis quam longin-
quitatem viae imputabant.” Horace shows his geographical knowledge
by saying : “ Not those who drink of the deep Danube shall now break
the Julian edicts; not the Getae, not the Seres, nor the perfidious
Persians, nor those born on the river Tanals.” [Odes, Bk. IY, Ode
15, 21-24.]

3 “ Qua virtutis moderationisque f ama Indos etiam ac Scythas auditu
modo cognitos pellexit ad amicitiam suam populique Romani ultra per
legatos petendam.” [Reinaud, loc. cit., p. 180.]

4 Reinaud, loc. cit., p. 180.

6 Georgies , II, 170-172. So Propertius {Elegies, III, 4):

Arina deus Caesar dites meditatur ad Indos
Et freta gemmiferi findere classe maris.

“ The divine Cresar meditated carrying arms against opulent India, and
with liis ships to cut the gem-bearing seas.”

6 Ileyd, loc. cit., Vol. I, p. 4.

7 Reinaud, loc. cit., p. 393.

THE BOETIIIUS QUESTION

number system of India, as several writers have main-
tained.1

Returning to the East, there are many evidences of the
spread of knowledge in and about India itself. In the
third century is.c. Buddhism began to be a connecting
medium of thought. It had already permeated the Hima-
laya territory, had reached eastern Turkestan, and had
probably gone thence to China. Some centuries later (in
62 a.d.) the Chinese emperor sent an ambassador to
India, and in 67 a.d. a Buddhist monk was invited to
China.2 Then, too, in India itself Asoka, whose name
has already been mentioned in this work, extended the
boundaries of his domains even into Afghanistan, so that
it was entirely possible for the numerals of the Punjab
to have worked their way north even at that early date.3

Furthermore, the influence of Persia must not be for-
gotten in considering this transmission of knowledge. In
the fifth century the Persian medical school at Jondi-
Sapur admitted both the Hindu and the Greek doctrines,
and FirdusI tells us that during the brilliant reign of

1 The title page of Calandri (1491), for example, represents Pythago-
ras with these numerals before him. [Smith, Rara Arithmetica, p. 46.]
Isaacus Yossius, Observationes ad Pomponium Melam de situ orbis, 1658,
maintained that the Arabs derived these numerals from the west. A
learned dissertation to this effect, but deriving them from the Romans
instead of the Greeks, was written by Ginanni in 1753 (Dissertatio
matliematica critica de numeralium notarum minuscularum origine, Ven-
ice, 1753). See also Mannert, De numerorum quos arabicos vocant vera
origine Pythagorica, Niirnberg, 1801. Even as late as 1827 Romagnosi
(in his supplement to Ricerche storiche suit' India etc., by Robertson,
Vol. II, p. 580, 1827) asserted that Pythagoras originated them. [R.
Bombelli, L'antica numerazione italica , Rome, 1876, p. 59.] Gow (Hist,
of Greek Math., p. 98) thinks that Iamblichus must have known a simi-
lar system in order to have worked out certain of his theorems, but
this is an unwarranted deduction from the passage given.

2 A. Hillebrandt, Alt-Indien, p. 179.

3 J. C. Marshman, loc. cit., chaps, i and ii.

THE HINDU- ARABIC NUMERALS

Khosru I,1 the golden age of Pahlavl literature, the
Hindu game of chess was introduced into Persia, at a
time when wars with the Greeks were bringing prestige
to the Sassanid dynasty.

Again, not far from the time of Boethius, in the sixth
century, the Egyptian monk Cosmas, in his earlier years
as a trader, made journeys to Abyssinia and even to
India and Ceylon, receiving the name Inclicopleustes (the
Indian traveler). His map (547 a.d.) shows some knowl-
edge of the earth from the Atlantic to India. Such a
man would, with hardly a doubt, have observed every
numeral system used by the people with whom he so-
journed,2 and whether or not he recorded his studies in
permanent form he would have transmitted such scraps
of knowledge by word of mouth.

As to the Arabs, it is a mistake to feel that their activi-
ties began with Mohammed. Commerce had always been
held in honor by them, and the Qoreish 3 had annually
for many generations sent caravans bearing the spices and
textiles of Yemen to the shores of the Mediterranean. In
the fifth century they traded by sea with India and even
with China, and Hira was an emporium for the wares of
the East,4 so that any numeral system of any part of the
trading world could hardly have remained isolated.

Long before the warlike activity of the Arabs, Alex-
andria had become the great market-place of the world.
From this center caravans traversed Arabia to Hadra-
maut, where they met ships from India. Others went
north to Damascus, while still others made then’ way

1 He reigned 531-579 a.d.; called Nu6irwan, the holy one.

2 J. Keane, The Evolution of Geography, London, 1899, p. 38.

3 The Arabs who lived in and about Mecca.

4 S. Guyard, in Encyc. Brit., 9th ed., Vol. XVI, p. 597-

TIIE BOETHIUS QUESTION

along the southern shores of the Mediterranean. Ships
sailed from the isthmus of Suez to all the commercial
ports of Southern Europe and up into the Black Sea.
Hindus were found among the merchants1 who fre-
quented the bazaars of Alexandria, and Brahmins were
reported even in Byzantium.

Such is a very brief resume of the evidence showing
that the numerals of the Punjab and of other parts of
India as well, and indeed those of China and farther
Persia, of Ceylon and the Malay peninsula, might well
have been known to the merchants of Alexandria, and
even to those of any other seaport of the Mediterranean,
in the time of Boethius. The Brahml numerals would
not have attracted the attention of scholars, for they had
no zero so far as we know, and therefore they were no
better and no worse than those of dozens of other sys-
tems. If Boethius was attracted to them it was probably
exactly as any one is naturally attracted to the bizarre
or the mystic, and he would have mentioned them in his
works only incidentally, as indeed they are mentioned in
the manuscripts in which they occur.

In answer therefore to the second question, Could
Boethius have known the Hindu numerals ? the reply
must be, without the slightest doubt, that he could easily
have known them, and that it would have been strange
if a man of his inquiring mind did not pick up many
curious bits of information of this kind even though he
never thought of making use of them.

Let us now consider the third question, Is there any
positive or strong circumstantial evidence that Boethius
did know these numerals ? The question is not new,

1 Oppert, loc. cit., p. 29.

THE IIINDU-ARABIC NUMERALS

nor is it much nearer being answered than it was over
two centuries ago when Wallis (1693) expressed his
doubts about it1 soon after Vossius (1658) had called
attention to the matter.2 Stated briefly, there are three
works on mathematics attributed to Boethius : 3 (1) the
arithmetic, (2) a work on music, and (3) the geometry.4

The genuineness of the arithmetic and the treatise on
music is generally recognized, but the geometry, which
contains the Hindu numerals with the zero, is under
suspicion.5 There are plenty of supporters of the idea
that Boethius knew the numerals and included them in
this book,6 and on the other hand there are as many who

1 “At non credendum est id in Autographis contigisse, aut vetusti-
oribus Codd. MSS.” [Wallis, Opera omnia , Vol. II, p. 11.]

2 In Observationes ad Pomponium Melam de situ orbis. The ques-
tion was next taken up in a large way by Weidler, loc. cit., I)e charac-
teribus etc., 1727, and in Spicilegium etc., 1755.

8 The best edition of these works is that of G. Friedlein, Anicii
Manlii Torquati Severini Boetii de institutione arithmetica libri duo , de
institutione musica libri quinque. Accedit geometria quae fertur Boetii.
. . . Leipzig. . . . mdccclxvii.

4 See also P. Tannery, “ Notes sur la pseudo-g6omdtrie de Bofece,”
in Bibliotheca Mathematical Yol. I (3), p. 39. This is not the geometry
in two books in which are mentioned the numerals. There is a manu-
script of this pseudo-geometry of the ninth century, but the earliest
one of the other work is of the eleventh century (Tannery), unless
the Vatican codex is of the tenth century as Friedlein (p. 372) asserts.

6 Friedlein feels that it is partly spurious, but he says: “Eorum
librorum, quos Boetius de geometria scripsisse dicitur, investigare
veram inscriptionem nihil aliud esset nisi operam et tempus perdere.”
[Preface, p. v.] N. Bubnov in the Russian Journal of the Ministry of
Public Instruction , 1907, in an article of which a synopsis is given in
the Jahrbucli iiber die Fortschritte der Mathematik for 1907, asserts that
the geometry was written in the eleventh century.

6 The most noteworthy of these was for a long time Cantor (Ge-
schiclite , Vol. I., 3d ed., pp. 587-588), who in his earlier days even
believed that Pythagoras had known them. Cantor says (Lie romischen
Agrimensoren , Leipzig, 1875, p. 130): “Uns also, wir wiederliolen es,
ist die Geometrie des Boetius edit, dieselbe Sclirift, welclie er nacli
Euklid bearbeitete, von welclier ein Codex bereits in Jahre 821 im

THE BOETHIUS QUESTION

feel that the geometry, or at least the part mentioning
the numerals, is spurious.1 The argument of those who
deny the authenticity of the particular passage in ques-
tion may briefly be stated thus :

1. The falsification of texts has always been the sub-
ject of complaint. It was so with the Romans,2 it was com-
mon in the Middle Ages,3 and it is much more prevalent

Kloster Reichenau vorhanden war, von welcker ein anderes Exemplar
im Jalire 982 zu Mantua in die Hande Gerbert’ s gelangte, von welcker
mannigfacke Ilandsckriften nock lieute vorkanden sind.” But against
tkis opinion of tlie antiquity of MSS. containing tkese numerals is
tke important statement of P. Tannery, perkaps tke most critical of
modern liistorians of matliematics, tkat none exists earlier tkan tke
eleventk century. See also J. L. Heiberg in Philologus, Zeitschrift f.
d. Iclass. Altertum, Yol. XLIII, p. 508.

Of Cantor’s predecessors, Th. II. Martin was one of tke most promi-
nent, kis argument for autlienticity appearing in tke Revue ArcMolo-
gique for 1856-1857, and in kis treatise Les signes numeraux etc.
See also M. Ckasles, “De la connaissance qu’ont eu les anciens d’une
numeration d^cimale 6crite qui fait usage de neuf cliiffres prenant
les valeurs de position,” Comptes rendus, Vol. VI, pp. 678-680; “Sur
l’origine de notre systfeme de numeration,” Comptes rendus , Vol.
VIII, pp. 72-81 ; and note “ Sur le passage du premier livre de la g6o-
metrie de Bofece, relatif k un nouveau systfeme de numeration,” in kis
work Apergu historique sur Vorigine et le devdloppement des methodes en
g6om£trie, of wliick tke first edition appeared in 1837.

1 J. L. Heiberg places tke book in the eleventh century on philo-
logical grounds, Philologus, loc. cit.; Woepcke, in Propagation, p. 44;
Blume, Lachmann, and Rudorff, Die Schriften der romischen Feldmesser,
Berlin, 1848 ; Boeckli, De abaco graecorum, Berlin, 1841 ; Friedlein,
in kis Leipzig edition of 1867; Weissenborn, Abhandlungen , Vol. II,
p. 185, kis Gerbert, pp. 1, 247, and his Geschichte der Einfiihrung der
jetzigen Ziffern in Europa durch Gerbert, Berlin, 1892, p. 11 ; Bayley,
loc. cit., p. 59 ; Gerliardt, deludes, p. 17, Entstehung und A usbreiiung,
p. 14 ; Nagl, Gerbert, p. 57 ; Bubnov, loc. cit. See also tke discussion
by Ckasles, Halliwell, and Libri, in the Comptes rendus, 1839, Vol. IX,
p. 447, and in Vols. VIII, XVI, XVII of tke same journal.

2 J. Marquardt, La vie privee des Romains, Vol. II (French trans.),
p. 505, Paris, 1893.

3 In a Plimpton manuscript of tke arithmetic of Boethius of tke thir-
teenth century, for example, tke Roman numerals are all replaced by
tke Arabic, and tke same is true in the first printed edition of tke book.

THE IIINDU-ARABIC NUMERALS

to-day than we commonly think. We have but to see
how every hymn-book compiler feels himself author-
ized to change at will the classics of our language, and
how unknown editors have mutilated Shakespeare, to see
how much more easy it was for medieval scribes to insert
or eliminate paragraphs without any protest from critics.1

2. If Boethius had known these numerals he would have
mentioned them in his arithmetic, but he does not do so.2

8. If he had known them, and had mentioned them in
any of his works, his contemporaries, disciples, and suc-
cessors would have known and mentioned them. But
neither Capella (c. 475)3 nor any of the numerous medi-
eval writers who knew the works of Boethius makes any
reference to the system.4

(See Smith’s Rara Arithmetical pp.434, 25-27.) D. E. Smith also cop-
ied from a manuscript of the arithmetic in the Laurentian library at

Florence, of 1370, the following forms, / 7 1 ■* < \c in»
which, of course, are interpolations. An interesting example of a for-
gery in ecclesiastical matters is in the charter said to have been given
by St. Patrick, granting indulgences to the benefactors of Glastonbury,
dated “In nomine domini nostri Jhesu Christi Ego Patricius liumilis
servunculus Dei anno incarnationis ejusdem ccccxxx.” Now if the
Benedictines are right in saying that Dionysius Exiguus, a Scythian
monk, first arranged the Christian chronology c. 532 a.d., this can
hardly be other than spurious. See Arbuthnot, loc. cit., p. 38.

1 Halliwell, in his Rara Mathematical p. 107, states that the disputed
passage is not in a manuscript belonging to Mr. Ames, nor in one at
Trinity College. See also Woepcke, in Propagation , pp. 37 and 42.
It was the evident corruption of the texts in such editions of Boethius
as those of Venice, 1490, Basel, 1540 and 1570, that led Woepcke
to publish his work Sur V introduction de V arithnietique indienne en
Occident.

2 They are found in none of the very ancient manuscripts, as, for
example, in the ninth-century (?) codex in the Laurentian library
which one of the authors has examined. It should be said, however,
that the disputed passage was written after the arithmetic, for it con-
tains a reference to that work. See the Eriedlein ed., p. 397.

8 Smith, Rara Arithmetical p. 66.

4 J. L. Heiberg, Philologus, Vol. XLIII, p. 507.

THE BOETHIUS QUESTION

4. The passage in question has all the appearance of
an interpolation by some scribe. Boethius is speaking of
angles, in his work on geometry, when the text suddenly
changes to a discussion of classes of numbers.1 This is
followed by a chapter in explanation of the abacus,2 in
which are described those numeral forms which are called
apices or caracteres .3 The forms 4 of these characters vary
in different manuscripts, but in general are about as
shown on page 88. They are commonly written with
the 9 at the left, decreasing to the unit at the right, nu-
merous writers stating that this was because they were
derived from Semitic sources in which the direction of
writing is the opposite of our own. This practice con-
tinued until the sixteenth century.5 6 The writer then
leaves the subject entirely, using the Roman numerals

1 “Nosse autem hums artis dispicientem, quid sint digiti, quid arti-
culi, quid compositi, quid incompositi numeri.” [Friedlein ed., p.395.]

2 l)e ratione abaci. In this he describes “ quandam formulam, quam
ob honorem sui praeceptoris mensam Pythagoream nominabant . . .
a posterioribus appellabatur abacus.” This, as pictured in the text, is
the common Gerbert abacus. In the edition in Migne’s Patrologia
Latina , Vol. LXIII, an ordinary multiplication table (sometimes called
Pythagorean abacus) is given in the illustration.

8 “ Habebant enim diverse formatos apices vel caracteres.” See the
reference to Gerbert on p. 117.

4 C. Henry, “Sur l’origine de quelques notations mathdmatiques,”

Itevue ArcMologique, 1879, derives these from the initial letters used as
abbreviations for the names of the numerals, a theory that finds few
supporters.

6 E.g., it appears in Schonerus, Algorilhmus Demonstrates, Nurn-
berg, 1534, f.A4. In England it appeared in the earliest English
arithmetical manuscript known, The Crafte of Nombrynge : “Iffforther-
more ye most vndirstonde that in this craft ben vsid teen figurys, as
here bene writen for ensampul, o 9 8 A 6 q g 3 2 1 . . . in the quyeh we
vse teen figurys of Inde. Questio. If why ten fyguris of Inde ? Solu-
cio. for as I have sayd afore thei were fonde fyrst in Inde of a kynge
of that Cuntre, that was called Algor.” See Smith, An Early English
Algorism , loc. cit.

TIIE IIINDU-ARABIC NUMERALS

Foums of the Numerals, Largely from Works on

the Abacus 1

1 2345 6 789 0

a I

b T

■S 9*

c 1

( s

d-l

e X

8 1

rh %

‘1

9 <s>

'S’

A f&

5 <S>

a Friedlein ed., p. 397. b Carlsruhe codex of Gerlando.

c Munich codex of Gerlando. d Carlsruhe codex of Bernelinus.

e Munich codex of Bernelinus. f Turchill, c. 1200.
s Anon. MS., thirteenth century, Alexandrian Library, Rome.
h Twelfth-century Boethius, Friedlein, p. 39G.

1 Vatican codex, tenth century, Boethius.

1 a, h, ', are from the Friedlein ed. ; the original in the manuscript
from which a is taken contains a zero symbol, as do all of the six
plates given by Friedlein. b-e from the Boncompagni Bulletino, Vol.
X, p. 596 ; f ibid., Vol. XV, p. 136 ; eMemorie della classe di sci., Reale
Ace. dei Lincei, An. CCLXXIV (1876-1877), April, 1877. A twelftli-
centuiy arithmetician, possibly John of Luna (Hispalensis, of Seville,
c. 1150), speaks of the great diversity of these forms even in his day,
saying: “Est autem in aliquibus figuram istarum apud multos diuer-
sitas. Qixidam enim septimam lianc figuram represent, ant .(f alii
autem sic uel sic -'J . Quidam vero quartain sic .” [Boncom-
pagni, Trattati , Vol. II, p. 28.]

THE BOETIIIUS QUESTION

for the rest of his discussion, a proceeding so foreign to
the method of Boethius as to be inexplicable on the
hypothesis of authenticity. Why should such a scholarly
writer have given them with no mention of their origin
or use ? Either he would have mentioned some histor-
ical interest attaching to them, or he would have used
them in some discussion ; he certainly would not have
left the passage as it is.

Sir E. Clive Bayley has added 1 a further reason for
believing them spurious, namely that the 4 is not of the
Nana Ghat type, but of the Kabul form which the Arabs
did not receive until 776 ; 2 so that it is not likely, even
if the characters were known in Europe hi the time of
Boethius, that this particular form was recognized. It
is worthy of mention, also, that in the six abacus forms
from the chief manuscripts as given by Friedlem,3 each
contains some form of zero, which symbol probably origi-
nated in India about this time or later. It could hardly
have reached Europe so soon.

As to the fourth question, Did Boethius probably know
the numerals ? It seems to be a fair conclusion, accord-
ing to our present evidence, that (1) Boethius might
very easily have known these numerals without the zero,
but, (2) there is no reliable evidence that he did know
them. And just as Boethius might have come in contact
with them, so any other inquiring mind might have done
so either in his time or at any time before they definitely
appeared in the tenth century. These centuries, five in
number, represented the darkest of the Dark Ages, and
even if these numerals were occasionally met and studied,
no trace of them would be likely to show itself in the

1 Loc. cit., p. 59. 2 Ibid., p. 101. 8 Loc. cit., p. 396.

THE IIINDU-ARABTC NUMERALS

literature of the period, unless by chance it should get
into the writings of some man like Alcuin. As a matter
of fact, it was not until the ninth or tenth century that
there is any tangible evidence of their presence in Chris-
tendom. They were probably known to merchants here
and there, but in their incomplete state they were not of
sufficient importance to attract any considerable attention.

As a result of this brief survey of the evidence several
conclusions seem reasonable: (1) commerce, and travel
for travel’s sake, never died out between the East and the
West; (2) merchants had every opportunity of knowing,
and would have been unreasonably stupid if they had
not known, the elementary number systems of the peo-
ples with whom they were trading, but they would not
have put this knowledge in permanent written form ;
(3) wandering scholars would have known many and
strange things about the peoples they met, but they too
were not, as a class, writers ; (4) there is every reason
a priori for believing that the gobar numerals would
have been known to merchants, and probably to some of
the wandering scholars, long before the Arabs conquered
northern Africa ; (5) the wonder is not that the Hindu-
Arabic numerals were known about 1000 A.D., and that
they were the subject of an elaborate work in 1202 by
Fibonacci, but rather that more extended manuscript evi-
dence of their appearance before that time has not been
found. That they were more or less known early in the
Middle Ages, certainly to many merchants of Christian
Europe, and probably to several scholars, but without
the zero, is hardly to be doubted. The lack of docu-
mentary evidence is not at all strange, in view of all
of the circumstances.

CHAPTER VI

THE DEVELOPMENT OF THE NUMERALS
AMONG THE ARABS

If the numerals had their origin in India, as seems
most probable, when did the Arabs come to know of
them ? It is customary to say that it was due to the in-
fluence of Mohammedanism that learning spread through
Persia and Arabia; and so it was, in part. But learning
was already respected in these countries long before Mo-
hammed appeared, and commerce flourished all through
this region. In Persia, for example, the reign of Khosru
Nuslrwan,1 the great contemporary of Justinian the law-
maker, was characterized not only by an improvement in
social and economic conditions, but by the cultivation of
letters. Khosru fostered learning, inviting to his court
scholars from Greece, and encouraging the introduction
of culture from the West as well as from the East. At
this time Aristotle and Plato were translated, and por-
tions of the Ilito-padesa, or Fables of Pilpay, were ren-
dered from the Sanskrit into Persian. All this means
that some three centuries before the great intellectual
ascendancy of Bagdad a similar fostering of learning was
taking place in Persia, and under pre-Mohanunedan
influences.

1 Khosru I, who began to reign in 531 a.d. See W. S. W. Vaux,
Persia, London, 1875, p. 109; Th. Noldeke, Aufsatze zur persischen
Geschichte, Leipzig, 1887, p. 113, and his article in the ninth edition
of the Encyclopedia Britannica.

TIIE IIINDU-AllABIC NUMERALS

The first definite trace that we have of the introduc-
tion of the Hindu system into Arabia dates from 773 a.d.,1
when an Indian astronomer visited the court of the ca-
liph, bringing with him astronomical tables which at the
caliph’s command were translated into Arabic by Al-
Fazari.2 Al-KhowarazmI and Habash (Ahmed ibn 'Ab-
dallah, died c. 870) based their well-known tables upon
the work of Al-Fazari. It may be asserted as highly
probable that the numerals came at the same time as the
tables. They were certainly known a few decades later,
and before 825a.d., about which time the original of the
Algoritmi de numero Indorum was written, as that work
makes no pretense of being the first work to treat of the
Hindu numerals.

The three writers mentioned cover the period from the
end of the eighth to the end of the ninth century. While
the historians Al-Mas'udI and Al-Blruni follow quite
closely upon the men mentioned, it is well to note again
the Arab writers on Hindu arithmetic, contemporary with
Al-Ivhowarazmi, who were mentioned in chapter I, viz.
Al-Kindi, Sened ibn 'All, and Al-Sufi.

For over five hundred years Arabic writers and others
continued to apply to works on arithmetic the name
“ Indian.” In the tenth century such writers are 'Abdal-
lah ibn al-Hasan, Abu ’l-Qasim3 (died 987 a.d.) of An-
tioch, and Mohammed ibn 'Abdallah, Abu Nasr 4 (c. 982),
of Ivalwada near Bagdad. Others of the same period or

1 Colebrooke, Essays , Vol. II, p. 504, on the authority of Ibn al-
Adami, astronomer, in a work published by his continuator Al-Qasim
in 920 a. d.; Al-BIrunI, India, Vol. II, p. 15.

2 H. Suter, Die Mathematiker etc., pp. 4-5, states that Al-Fazari
died between 796 and 806.

8 Suter, loc. cit., p. 63.

4 Suter, loc. cit., p. 74.

DEVELOPMENT OF THE NUMERALS

earlier (since they are mentioned in the Fihrist } 987 A.D.),
who explicitly use the word “Hindu” or “Indian,” are
Sinan ibn al-Fath2 of Harran, and Ahmed ibn 'Omar,
al-Karabls!.8 In the eleventh century come Al-BIruni4
(973-1048) and 'Ali ibn Ahmed, Abu ’1-Hasan, Al-
Nasawl6 (c. 1030). The following century brings simi-
lar works by Ishaq ibn Yusuf al-Sardafi6 and Samu’il
ibn Yahya ibn 'Abbas al-Magrebl al-AndalusI7 (c. 1174),
and in the thirteenth century are 'Abdallatlf ibn Yusuf
ibn Mohammed, Muwaffaq al-Dln Abu Mohammed al-
Bagdadi8 (c. 1231), and Ibn al-Banna.9

The Greek monk Maximus Planudes, writing in the
first half of the fourteenth century, followed the Arabic
usage in call i no- his work Indian Arithmetic.10 There were
numerous other Arabic writers upon arithmetic, as that
subject occupied one of the high places among the sciences,
but most of them did not feel it necessary to refer to the
origin of the symbols, the knowledge of which might well
have been taken for granted.

1 Suter, Das Mathematiker-Yerzeichniss im Fihrist. The references
to Suter, unless otherwise stated, are to his later work Die Mathemati-
ker und Astronomen der Araber etc.

2 Suter, Fihrist, p. 37, no date.

8 Suter, Fihrist, p. 38, no date.

4 Possibly late tenth, since he refers to one arithmetical work which
is entitled Book of the Cyphers in his Chronoldgy, English ed., p. 132.
Suter, Die Mathematiker etc., pp. 98-100, does not mention this work ;
see the Nachtrdge und Berichtigungen , pp. 170-172.

6 Suter, pp. 96-97.

6 Suter, p. 111.

7 Suter, p. 124. As the name shows, he came from the West.

8 Suter, p. 138.

9 Ilankel, Zur Geschichte der Mathematik , p. 256, refers to him as
writing on the Hindu art of reckoning; Suter, p. 162.

10 'Pt ifocpopla kclt 'Ivdotis, Greek ed., C. I. Gerhardt, Halle, 1865 ;
and German translation, Das Rechenbuch dcs Maximus Flanudes, H.
Waschke, Halle, 1878.

TIIE HINDU-ARABIC NUMERALS

One document, cited by Woepcke,1 is of special inter-
est since it shows at an early period, 970 a.d., the use
of the ordinary Arabic forms alongside the gobar. The
title of the work is Interesting and Beautiful Problems on
Numbers copied by Ahmed ibn Mohammed ibn 'Abdaljalll,
Abu Sa'Id, al-SijzI,2 (951-1024) from a work by a priest
and physician, Nazlf ibn Yumn,3 al-Qass (died c. 990).
Suter does not mention this work of Nazlf.

The second reason for not ascribing too much credit *
to the purely Arab influence is that the Arab by himself
never showed any intellectual strength. What took place
after Mohammed had lighted the fire in the hearts of his
people was just what always takes place when different
types of strong races blend, — a great renaissance in
divers lines. It was seen in the blending of such types at
Miletus in the time of Thales, at Rome in the days of
the early invaders, at Alexandria when the Greek set
firm foot on Egyptian soil, and we see it now when all
the nations mingle their vitality in the New World. So
when the Arab culture joined with the Persian, a new
civilization rose and flourished.4 The Arab influence
came not from its purity, but from its intermingling with
an influence more cultured if less virile.

As a result of this interactivity among peoples of diverse
interests and powers, Mohammedanism was to the world
from the eighth to the thirteenth century what Rome and
Athens and the Italo-Hellenic influence generally had

1 “Sur une donn4e historique relative & l’emploi des chiffres in-
diens par les Arabes,” Tortolini’s Annali di scienze mat. efis., 1855.

2 Suter, p. 80.

8 Suter, p. 68.

■* Sprenger also calls attention to this fact, in the Zeitschrift d.
deutschen monjenldnd. Gesellschaft , Vol. XLV, p. 367.

DEVELOPMENT OF THE NUMERALS

been to the ancient civilization. “ If they did not possess
the spirit of invention which distinguished the Greeks
and the Hindus, if they did not show the p'erseverance
in their observations that characterized the Chinese
astronomers, they at least possessed the virility of a new
and victorious people, with a desire to understand what
others had accomplished, and a taste which led them
with equal ardor to the study of algebra and of poetry,
of philosophy and of language.” 1

It was in 622 a.d. that Mohammed fled from Mecca,
and within a century from that time the crescent had
replaced the cross in Christian Asia, in Northern Africa,
and in a goodly portion of Spain. The Arab empire was
an ellipse of learning with its foci at Bagdad and Cor-
dova, and its rulers not infrequently took pride in de-
manding intellectual rather than commercial treasure as
the result of conquest.2

It was under these influences, either pre-Mohammedan
or later, that the Hindu numerals found their way to the
North. If they were known before Mohammed’s time,
the proof of this fact is now lost. This much, however,
is known, that in the eighth century they were taken to
Bagdad. It was early in that century that the Moham-
medans obtained their first foothold in northern India,
thus foreshadowing an epoch of supremacy that endured
with varied fortunes until after the golden age of Akbar
the Great (1542-1605) and Shah Jehan. They also con-
quered Khorassan and Afghanistan, so that the learning
and the commercial customs of India at once found easy

1 Libri, Histoire des matMmatiques , Vol. I, p. 147.

2 “Dictant la paix & l’empereur de Constantinople, 1’ Arabs vieto-
rieux demandait des manuscrits et des savans.” [Libri, loc. cit.,

p. 108.]

TIIE IIINDU-AKABIC NUMERALS

access to the newly-established schools and the bazaars of
Mesopotamia and western Asia. The particular paths of
conquest and of commerce were either by way of the
Khyber Pass and through Kabul, Herat and Khorassan,
or by sea through the strait of Ormuz to Basra (Busra)
at the head of the Persian Gulf, and thence to Bagdad.
As a matter of fact, one form of Arabic numerals, the one
now in use by the Arabs, is attributed to the influence of
Kabul, while the other, which eventually became our nu-
merals, may very likely have reached Arabia by the other
route. It is in Bagdad,1 Dar al-Salam — “the Abode of
Peace,” that our special interest in the introduction of the
numerals centers. Built upon the ruins of an ancient
town by Al-Mansur2 in the second half of the eighth
century, it lies in one of those regions where the converg-
ing routes of trade give rise to large cities.3 Quite as
well of Bagdad as of Athens might Cardinal Newman
have said : 4

“What it lost in conveniences of approach, it gained
in its neighborhood to the traditions of the mysterious
East, and in the loveliness of the region in which it lay.
Hither, then, as to a sort of ideal land, where all arche-
types of the great and the fair were found in substantial
being, and all departments of truth explored, and all
diversities of intellectual power exhibited, where taste
and philosophy were majestically enthroned as in a royal
court, where there was no sovereignty but that of mind,
and no nobility but that of genius, where professors were

1 Persian bagadata , “God-given.”

2 One of the Abbassides, the (at least pretended) descendants of
'A 1- Abbas, uncle and adviser of Mohammed.

:1 E. Reclus, Asia, American ed., N. Y., 1891, Vol. IV, p. 227.

i Historical Sketches, Vol. Ill, chap, iii,

DEVELOPMENT OF THE NUMERALS

rulers, and princes did homage, thither flocked continually
from the very corners of the orbis terrarum the many-
tongued generation, just rising, or just risen into man-
hood, in order to gain wisdom.” F,pr here it was that
Al-Mansur and Al-Mamun and Harun al-Rashld (Aaron
the Just) made for a time the world’s center of intellec-
tual activity in general and in the domain of mathematics
in particular.1 It was just after the Sindliind was brought
to Bagdad that Mohammed ibn Musa al-Khowarazmi,
whose name has already been mentioned,2 was called to
that city. He was the most celebrated mathematician of
his time, either in the East or W est, writing treatises on
arithmetic, the sundial, the astrolabe, chronology, geom-
etry, and algebra, and giving through the Lathi translit-
eration of his name, algoritmi, the name of algorism to the
early arithmetics using the new Hindu numerals.3 Appre-
ciating at once the value of the position system so recently
brought from India, he wrote an arithmetic based upon
these numerals, and this was translated into Latin in the
time of Adelhard of Bath (c. 1130), although possibly by
his contemporary countryman Robert Cestrensis.4 This
translation was found in Cambridge and was published
by Boncompagni in 1857.5 6

Contemporary with Al-Kliowarazml, and working also
under Al-Mamun, was a Jewish astronomer, Abu ’1-Teiyib,

1 On its prominence at that period see Villicus, p. 70.

2 See pp. 4-5.

8 Smith, D. E., in the Cantor Festschrift , 1909, note pp. 10-11. See
also F. Woepcke, Propagation.

4 Enestrom, in Bibliotheca Mathematical Vol. I (3), p. 499 ; Cantor,

Geschichte, Vol. 1(3), p. 671.

6 Cited in Chapter I. It begins: “Dixit algoritmi : laudes deo rec-
tori nostro atque defensori dicamus dignas.” It is devoted entirely
to the fundamental operations and contains no applications,

THE IIINDU-AEABIC NUMERALS

Senecl ibn rAli, who is said to have adopted the Moham-
medan religion at the caliph’s request. He also wrote a
work on Hindu arithmetic,1 so that the subject must have
been attracting considerable attention at that time. In-
deed, the struggle to have the Hindu numerals replace
the Arabic did not cease for a long time thereafter. 'All
ibn Alnned al-NasawI, in his arithmetic of c. 1025, tells
us that the symbolism of number was still unsettled in
his day, although most people preferred the strictly ,
Arabic forms.2

W e thus have the numerals in Arabia, in two forms :
one the form now used there, and the other the one used
by Al-Khowarazmi. The question then remains, how did
this second form find its way into Europe ? and this ques-
tion will be considered in the next chapter.

1 M. Steinschneider, “Die Mathematik bei den Juden,” Bibliotheca
Mathematica , Vol. VIII (2), p. 99. See also the reference to this writer
in Chapter I.

2 Part of this work has been translated from a Leyden MS. by P.
Woepcke, Propagation , and more recently by H. Suter, Bibliotheca
Mathematica, Yol. VII (3), pp. 113-119.

CHAPTER YI I

THE DEFINITE INTRODUCTION OF THE NUMERALS

INTO EUROPE

It being doubtful whether Boethius ever knew the
Hindu numeral forms, certainly without the zero in any
case, it becomes necessary now to consider the question
of their definite introduction into Europe. From what
lias been said of the trade relations between the East and
the West, and of the probability that it was the trader
rather than the scholar who carried these numerals from
their original habitat to various commercial centers, it is
evident that we shall never know when they first made
them inconspicuous entrance into Europe. Curious cus-
toms from the East and from the tropics, — concerning
games, social peculiarities, oddities of dress, and the like,
— are continually being related by sailors and traders in
their resorts in New York, London, Hamburg, and Rot-
terdam to-day, customs that no scholar has yet described
in print and that may not become known for many years,
if ever. And if this be so now, how much more would it
have been true a thousand years before the invention of
printing, when learning was at its lowest ebb. It was at
this period of low esteem of culture that the Hindu numer-
als undoubtedly made their first appearance in Europe.

There were many opportunities for such knowledge to
reach Spain and Italy. In the first place the Moors went
into Spain as helpers of a claimant of the throne, and

100

THE II1NDU-ARABIC NUMERALS

remained as conquerors. The power of the Goths, who
had held Spain for three centuries, was shattered at the
battle of Jerez de la Frontera in 711, and almost imme-
diately the Moors became masters of Spam and so re-
mained for five hundred years, and masters of Granada
for a much longer period. Until 850 the Christians were
absolutely free as to religion and as to holding political
office, so that priests and monks were not infrequently
skilled both in Latin and Arabic, acting as official trans-
lators, and naturally reporting directly or indirectly to
Rome. There was indeed at this time a complaint that
Christian youths cultivated too assiduously a love for
the literature of the Saracen, and married too frequently
the daughters of the infidel.1 It is true that this happy
state of affairs was not permanent, but while it lasted
the learning and the customs of the East must have be-
come more or less the property of Christian Spain. At
this time the gobar numerals were probably in that coun-
try, and these may well have made their way into Europe
from the schools of Cordova, Granada, and Toledo.

Furthermore, there was abundant opportunity for the
numerals of the East to reach Europe through the jour-
neys of travelers and ambassadors. It was from the rec-
ords of Suleiman the Merchant, a well-known Arab trader
of the ninth century, that part of the story of Sindbad
the Sailor was taken.2 Such a merchant would have been
particularly likely to know the numerals of the people
whom he met, and he is a type of man that may well have
taken such symbols to European markets. A little later,

1 A. Neander, General History of the Christian Religion a,nd Church ,
5th American ed., Boston, 1855, Vol. Ill, p. 335,

2 Beazley, loc. cit., Vol. I, p. 49.

DEFINITE INTRODUCTION INTO EUROPE 101

Abu T-Hasan 'All al-Mas'udi (cl. 956) of Bagdad traveled
to the China Sea on the east, at least as far south as
Zanzibar, and to the Atlantic on the west,1 and he speaks
of the nine figures with which the Hindus reckoned.2 3 4

There was also a Bagdad merchant, one Abu ’1-Qasim
'Obeidalliih ibn Ahmed, better known by his Persian
name Ibn Khordadbeh,8 who wrote about 850 a.d. a
work entitled Book of Roads and Provinces 4 in which the
following graphic account appears:5 “The Jewish mer-
chants speak Persian, Roman (Greek and Latin), Arabic,
French, Spanish, and Slavic. They travel from the West
to the East, and from the East to the West, sometimes
by land, sometimes by sea. They take ship from France
on the Western Sea, and they voyage to Farama (near
the ruins of the ancient Pelusium) ; there they transfer
their goods to caravans and go by land to Colzom (on the
Red Sea). They there reembark on the Oriental (Red)
Sea and go to Hejaz and to Jiddah, and thence to the
Sind, India, and China. Returning, they bring back the
products of the oriental lands. . . . These journeys are
also made by land. The merchants, leaving France and
Spain, cross to Tangier and thence pass through the
African provinces and Egypt. They then go to Ram-
leh, visit Damascus, Kufa, Bagdad, and Basra, penetrate
into Ahwaz, Fars, Kerman, Sind, and thus reach India
and China.” Such travelers, about 900 a.d., must neces-
sarily have spread abroad a knowledge of all number

1 Beazley, loc. cit., Yol. I, pp. 60, 460.

2 See pp. 7-8.

3 The name also appears as Mohammed Abu’ 1-Qasim, and Ibn Hau-
qal. Beazley, loc. cit., Vol. I, p. 45.

4 Kitab al-masalik wa' l-mamalik.

6 Reinaud, M6m. sur V Inde ; in Gerhardt, Rtud.es, p. 18.

102

THE HINDU- ARABIC NUMERALS

systems used in recording prices or in the computations
of the market. There is an interesting witness to this
movement, a cruciform brooch now in the British Mu-
seum. It is English, certainly as early as the eleventh
century, but it is inlaid with a piece of paste on which
is the Mohammedan inscription, in Ivufic characters,
“ There is no God but God.” How did such an inscrip-
tion find its way, perhaps in the time of Alcuin of York,
to England ? And if these Kufic characters reached-
there, then why not the numeral forms as well ?

Even in literature of the better class there appears
now and then some stray proof of the important fact
that the great trade routes to the far East were never
closed for long, and that the customs and marks of trade
endured from generation to generation. The Grulistan of
the Persian poet Sa'di1 contains such a passage:

“ I met a merchant who owned one hundred and forty
camels, and fifty slaves and porters. . . . He answered to
me : ‘ I want to carry sulphur of Persia to China, which
in that country, as I hear, bears a high price ; and thence
to take Chinese ware to Roum ; and from Roum to load
up with brocades for Hind ; and so to trade Indian steel
Qptilab') to Ilalib. From Halib I will convey its glass to
Yeman, and carry the painted cloths of Yeman back to
Persia.’ ” 2 On the other hand, these men were not of the
learned class, nor would they preserve in treatises any
knowledge that they might have, although this knowl-
edge would occasionally reach the ears of the learned as
bits of curious information.

1 Born at Shiraz in 1193. He himself had traveled from India
to Europe.

2 Gulistan ( Rose Garden ), Gateway the third, XXII. Sir Edwin
Arnold’s translation, N.Y., 1899, p. 177.

DEFINITE INTRODUCTION INTO EUROPE 103

There were also ambassadors passing back and forth
from time to time, between the East and the West, and
in particular during the period when these numerals
probably began to enter Europe. Thus Charlemagne
(c. 800) sent emissaries to Bagdad just at the time of
the opening of the mathematical activity there.1 And
with such ambassadors must have gone the adventurous
scholar, inspired, as Alcuin says of Archbishop Albert
of York (7(:>0— 780),2 to seek the learning of other lands.
Furthermore, the Nestorian communities, established in
Eastern Asia and in India at this time, were favored both
by the Persians and by then’ Mohammedan conquerors.
The Nestorian Patriarch of Syria, Timotheus (778-820),
sent missionaries both to India and to China, and a bishop
was appointed for the latter field. Ibn W ahab, who trav-
eled to China in the ninth century, found images of Christ
and the apostles in the Emperor’s court.3 Such a learned
body of men, knowing intimately the countries in which
they labored, could hardly have failed to make strange
customs known as they returned to their home stations.
Then, too, in Alfred’s time (849-901) emissaries went

1 Cunningham, loc. cit., p. 81.

2 Putnam, Books, Vol. I, p. 227 :

“ Non semel externas peregrino tramite terras
Jam peragravit ovans, sopliiae deductus amore,

Si quid forte novi librorum seu studiorum
Quod secuin ferret, terris reperiret in illis.

Hie quoque Romuleum venit devotus ad urbem.”

(“ More than once he has traveled joyfully through remote regions and
by strange roads, led on by his zeal for knowledge and seeking to discover
in foreign lands novelties in books or in studies which he could take back
with him. And this zealous student journeyed to the city of Romulus.”)

3 A. Neander, General History of the Christian Religion and Church ,
6th American ed., Boston, 1855, Vol. Ill, p. 89, note 4 ; Libri, Histoire ,
Vol. I, p. 143.

104

THE IIINDU-ARABIC NUMERALS

from England as far as India,1 and generally in the
Middle Ages groceries came to Europe from Asia as now
they come from the colonies and from America. Syria,
Asia Minor, and Cyprus furnished sugar and wool, and
India yielded her perfumes and spices, while rich tapes-
tries for the courts and the wealthy burghers came from
Persia and from China.2 Even in the time of Justinian
(c. 550) there seems to have been a silk trade with China,
which country in turn carried on commerce with Ceylon,3
and reached out to Turkestan where other merchants
transmitted the Eastern products westward. In the sev-
enth century there was a well-defined commerce between
Persia and India, as well as between Persia and Con-
stantinople.4 The Byzantine commerciarii were stationed
at the outposts not merely as customs officers but as
government purchasing agents.5

Occasionally there went along these routes of trade
men of real learning, and such would surely have carried
the knowledge of many customs back and forth. Thus
at a period when the numerals are known to have been
partly understood in Italy, at the opening of the eleventh
century, one Constantine, an African, traveled from Italy
through a great part of Africa and Asia, even on to
India, for the purpose of learning the sciences of the
Orient. He spent thirty-nine years in travel, having
been hospitably received in Babylon, and upon his return
he was welcomed with great honor at Salerno.6

A very interesting illustration of this intercourse also
appears in the tenth century, when the son of Otto I

1 Cunningham, loc. cit., p. 81. 4 Ibid., p. 21.

2 Heyd, loc. cit., Vol. I, p. 4. 5 Ibid., p. 23.

8 Ibid., p. 5. 0 Libri, Histoire , Vol. I, p. 167.

DEFINITE INTRODUCTION INTO EUROPE 105

(936-973) married a princess from Constantinople. This
monarch was in touch with the Moors of Spain and
invited to his court numerous scholars from abroad,1
and his intercourse with the East as well as the West
must have brought together much of the learning of
each.

Another powerful means for the circulation of mysti-
cism and philosophy, and more or less of culture, took its
start just before the conversion of Constantine (c. 312),
in the form of Christian pilgrim travel. This was a
feature peculiar to the zealots of early Christianity,
found in only a slight degree among their Jewish prede-
cessors in the annual pilgrimage to Jerusalem, and
almost wholly wanting in other pre-Christian peoples.
Chief among these early pilgrims were the two Placen-
tians, John and Antonine the Elder (c. 303), who, in
them wanderings to Jerusalem, seem to have started a
movement which culminated centuries later in the cru-
sades.2 In 333 a Bordeaux pilgrim compiled the first
Christian guide-book, the Itinerary from Bordeaux to
Jerusalem ,3 and from this time on the holy pilgrimage
never entirely ceased.

Still another certain route for the entrance of the nu-
merals into Christian Europe was through the pillaging
and trading carried on by the Arabs on the northern
shores of the Mediterranean. As early as 652 a.d., in
the thirtieth year of the Hejira, the Mohammedans de-
scended upon the shores of Sicily and took much spoil.
Hardly had the wretched Constans given place to the

1 Picavet, Gerbert , un pape philosopher d'aprcs Vhistoire et d'aprbs
la tegende, Paris, 1897, p. 19.

2 Beazley, loc. cit., Vol. I, cliap. i, and p. 54seq. 8 Ibid., p. 57.

106

TIIE IIINDU-AIIABIC NUMERALS

young Constantine IV when they again attacked the
island and plundered ancient Syracuse. Again in 827,
under Asad, they ravaged the coasts. Although at this
time they failed to conquer Syracuse, they soon held a
good part of the island, and a little later they success-
fully besieged the city. Before Syracuse fell, however,
they had plundered the shores of Italy, even to the walls
of Rome itself ; and had not Leo IV, in 849, repaired the
neglected fortifications, the effects of the Moslem raid of
that year might have been very far-reaching. Ibn Ivhor-
dadbeh, who left Bagdad hi the latter part of the ninth
century, gives a picture of the great commercial activity
at that time in the Saracen city of Palermo. In this same
century they had established themselves in Piedmont,
and in 906 they pillaged Turin.1 On the Sorrento pen-
insula the traveler who climbs the hill to the beautiful
Ravello sees still several traces of the Arab architecture,
reminding him of the fact that about 900 a.d. Amalfi was
a commercial center of the Moors.2 Not only at this time,
but even a century earlier, the artists of northern India
sold their wares at such centers, and in the courts both of
Harun al-Rashid and of Charlemagne.3 Thus the Arabs
dominated the Mediterranean Sea long before Venice

“ held the gorgeous East in fee
And was the safeguard of the West,”

and long before Genoa had become her powerful rival.4

1 Libri, Histone, Vol. I, p. 110, n., citing authorities, and p. 152.

2 Possibly the old tradition, “ Prima dedit nautis usum magnetis
Amalphis,” is true so far as it means the modern form of compass
card. See Beazley, loc. cit., Vol. II, p. 398.

3 R. C. Dutt, loc. cit., Vol. II, p. 312.

4 E. J. Payne, in The Cambridge Modern History , London, 1902,
Vol. I, chap. i.

DEFINITE INTRODUCTION INTO EUROPE 107

Only a little later than this the brothers Nicolo and
Maffeo Polo entered upon their famous wanderings.1
Leaving Constantinople in 1260, they went by the Sea
of Azov to Bokhara, and thence to the court of Ivublai
Khan, penetrating China, and returning by way of Acre
in 1269 with a commission which required them to go
back to China two years later. This time they took
with them Nicolo’s son Marco, the historian of the jour-
ney, and went across the plateau of Pamir; they spent
about twenty years in China, and came back by sea from
China to Persia.

The ventures of the Poli were not long unique, how-
ever: the thirteenth century had not closed before Roman
missionaries and the merchant Petrus de Lucolongo had
penetrated China. Before 1350 the company of mission-
aries was large, converts were numerous, churches and
Franciscan convents had been organized in the East,
travelers were appealing for the truth of their accounts
to the “many” persons in Venice who had been in China,
Tsuan-chau-fu had a European merchant community,
and Italian trade and travel to China was a thing that
occupied two chapters of a commercial handbook.2

1 Geo. Phillips, “ The Identity of Marco Polo’s Zaitun with Chang-
chau, in T’oung pao,” Archives pour servir a V etude de Vhistoire de
VAsie orientate, Leyden, 1890, Vol. I, p. 218. W. Heyd, Geschichte des
Levanthandels im Mittelalter, Yol. II, p. 216.

The Palazzo dei Poli, where Marco was born and died, still stands
in the Corte del Milione, in Venice. The best description of the Polo
travels, and of other travels of the later Middle Ages, is found in
C. R. Beazley’s Dawn of Modem Geography , Vol. Ill, chap, ii, and
Part II.

2 Heyd, loc. cit., Vol. II, p. 220 ; H. Yule, in Encyclopaedia Britan-
nica , 9th (10th) or 11th ed., article “China.” The handbook cited is
Pegolotti’s Libro di divisamenti di paesi, chapters i-ii, where it is im-
plied that 860,000 would be a likely amount for a merchant going to
China to invest in his trip.

108

THE IIINDU-ARABIC NUMERALS

It is therefore reasonable to conclude that in the Mid-
dle Ages, as in the time of Boethius, it was a simple
matter for any inquiring scholar to become acquainted
with such numerals of the Orient as merchants may have
used for warehouse or price marks. And the fact that
Gerbert seems to have known only the forms of the sim-
plest of these, not comprehending their full significance,
seems to prove that he picked them up in just this way.

Even if Gerbert did not bring his knowledge of the ,
Oriental numerals from Spam, he may easily have ob-
tained them from the marks on merchant’s goods, had he
been so inclined. Such knowledge was probably ob-
tainable in various parts of Italy, though as parts of mere
mercantile knowledge the forms might soon have been
lost, it needing the pen of the scholar to preserve them.
Trade at this time was not stagnant. During the eleventh
and twelfth centuries the Slavs, for example, had very
great commercial interests, their trade reaching to Kiev
and Novgorod, and thence to the East. Constantinople
was a great clearing-house of commerce with the Orient,1
and the Byzantine merchants must have been entirely
familiar with the various numerals of the Eastern peoples.
In the eleventh century the Italian town of Amalfi estab-
lished a factory2 in Constantinople, and had trade re-
lations with Antioch and Egypt. Venice, as early as the
ninth century, had a valuable trade with Syria and Cairo.3
Fifty years after Gerbert died, in the time of Cnut, the
Dane and the Norwegian pushed their commerce far be-
yond the northern seas, both by caravans through Russia
to the Orient, and by their venturesome barks which

1 Cunningham, loc. cit., p. 194. 2 I.e. a commission house.

3 Cunningham, loc. cit., p. 186.

DEFINITE INTRODUCTION INTO EUROPE 109

sailed through the Strait of Gibraltar into the Medi-
terranean.1 Only a little later, probably before 1200 A.D.,
a clerk hi the service of Thomas a Becket, present at the
latter’s death, wrote a life of the martyr, to which (fortu-
nately for our purposes) he prefixed a brief eulogy of
the city of London.2 This clerk, William Fitz Stephen
by name, thus speaks of the British capital :

Aurum mittit Arabs : species et thura Sab 02 us :

Arma Sythes : oleum palmarum divite sylva
Pingue solum Babylon: Nilus lapides pretiosos :

Norwegi, Russi, varium grisum, sabdinas :

Seres, purpureas vestes : Galli, sua vina.

Although, as a matter of fact, the Arabs had no gold to
send, and the Scythians no arms, and Egypt no precious
stones save only the turquoise, the Chinese (Seres) may
have sent their purple vestments, and the north her sables
and other furs, and France her wines. At any rate the
verses show very clearly an extensive foreign trade.

Then there were the Crusades, which in these times
brought the East in touch with the West. The spirit of
the Orient showed itself in the songs of the troubadours,
and the baudeldn ,3 the canopy of Bagdad,4 5 became com-
mon in the churches of Italy. I11 Sicily and in Venice
the textile industries of the East found place, and made
then* way even to the Scandinavian peninsula.6

W e therefore have this state of affairs : There was
abundant intercourse between the East and West for

1 J. R. Green, Short History of the English People, New York, 1890,

p. 66.

2 W. Besant, London, New York, 1892, p. 43.

8 Baldakin, baldekin, baldachino.

4 Italian Baldacco.

5 J. K. Mumford, Oriental Bugs, New York, 1901, p. 18.

110

T11E IIINDU-ARABIC NUMERALS

some centuries before the Hindu numerals appear in
any manuscripts in Christian Europe. The numerals
must of necessity have been known to many traders in
a country like Italy at least as early as the ninth century,
and probably even earlier, but there "was no reason for
preserving them in treatises. Therefore when a man like
Gerbert made them known to the scholarly circles, he
was merely describing what had been familiar hi a small
way to many people hi a different walk of life.

Since Gerbert 1 was for a long time thought to have
been the one to introduce the numerals into Italy,2 a
brief sketch of this unique character is proper. Born of
humble parents,3 this remarkable man became the coun-
selor and companion of kings, and finally wore the papal
tiara as Sylvester II, from 999 until his death hi 1003.4
He was early brought under the influence of the monks
at Aurillac, and particularly of Rahnund, who had been
a pupil of Odo of Cluny, and there in due time he him-
self took holy orders. He visited Spam hi about 967 hi
company with Count Borel,6 remaining there three years,

1 Or Girbert, the Latin forms Gerhertus and Girhertus appearing
indifferently in the documents of his time.

2 See, for example, J. C. Heilbronner, Historia matheseos universes ,
p. 740.

8 “ Obscuro loco natum,” as an old chronicle of Aurillac has it.

4 N. Bubnov, Gerberti posted Silvestn II papae opera mathematical
Berlin, 1899, is the most complete and reliable source of information ;
Picavet, loc. cit., Gerbert etc.; Olleris, (Euvres de Gerbert , Paris, 1867 ;
Ilavet, Lettres de Gerbert , Paris, 1889 ; H. Weissenborn, Gerbert; Bei-
trdge zur Kenntnis der Mathematik des Mittelalters , Berlin, 1888, and
Zur Geschichte der Einfiihrung der jetzigen Ziff'ern in Europa durch
Gerbert , Berlin, 1892; Budinger, Ueber Gerberts wissenschaftliche und
politische Stellung , Cassel, 1851 ; Richer, “ Historianun liber III,” in
Bubnov, loc. cit., pp. 376-381 ; Nagl, Gerbert und die Rechenkunst des
10. Jahrhunderts, Vienna, 1888.

5 Richer tells of the visit to Aurillac by Borel, a Spanish noble-
man, just as Gerbert was entering into young manhood. He relates

DEFINITE INTRODUCTION INTO EUROPE 111

and studying under Bishop Hatto of Vich,1 a city in the
province of Barcelona, 2 then entirely under Christian
rule. Indeed, all of Gerbert’s testimony is as to the in-
fluence of the Christian civilization upon his education.
Thus he speaks often of his study of Boethius,3 so that
if the latter knew the numerals Gerbert would have
learned them from him.4 If Gerbert had studied hi any
Moorish schools he would, under the decree of the emir
Hisliam (787-822), have been obliged to know Arabic,
which would have taken most of his three years in
Spain, and of which study we have not the slightest
hint in any of his letters.5 On the other hand, Barce-
lona was the only Christian province in immediate touch
with the Moorish civilization at that time.6 Further-
more we know that earlier in the same century King
Alonzo of Asturias (d. 910) confided the education of
his son Ordono to the Arab scholars of the court of the

how affectionately the abbot received him, asking if there were men in
Spain well versed in the arts. Upon Borel’s reply in the affirmative,
the abbot asked that one of his young men might accompany him upon
his return, that he might cany on his studies there.

1 Vicus Ausona. Hatto also appears as Atton and Hatton.

2 This is all that we know of his sojourn in Spain, and this comes
from his pupil Richer. The stories told by Adhemar of Cliabanois, an
apparently ignorant and certainly untrustworthy contemporary, of his
going to Cordova, are unsupported. (See e.g. Picavet, p. 34.) Never-
theless this testimony is still accepted: K. von Raumer, for example
( Geschichte der Pddagogik , 6th ed., 1890, Vol. I, p. 6), says “ Mathe-
matik studierte man im Mittelalter bei den Arabem in Spanien. Zu
ihnen gieng Gerbert, naclimaliger Pabst Sylvester II.”

8 Thus in a letter to Aldabex-on he says : “ Quos post repperimus
speretis, id est VIII volumina Boeti de astrologia, praeclarissima quoque
figurarum geometric, aliaque non minus admiranda ” (Epist. 8). Also
in a letter to Rainard (Epist. 130), he says : “Ex tuis sumptibus fac
ut michi scribantur M. Manlius (Manilius in one MS.) de astrologia.”

4 Picavet, loc. cit., p. 31.

6 Picavet, loc. cit., p. 36.

6 Havet, loc. cit., p. vii.

112

THE IIINDU-AIIABIC NUMERALS

wall of Saragossa,1 so that there was more or less of
friendly relation between Christian and Moor.

After his three years in Spam, Gerbert went to Italy,
about 970, where he met Pope John XIII, being by him
presented to the emperor Otto I. Two years later (972),
at the emperor’s request, he went to Rheims, where he
studied philosophy, assisting to make of that place an ed-
ucational center ; and in 983 he became abbot at Bobbio.
The next year he returned to Rheims, and became arch-
bishop of that diocese in 991. For political reasons he
returned to Italy in 996, became archbishop of Ravenna
hr 998, and the following year was elected to the papal
chan. Far ahead of his age in wisdom, he suffered as
many such scholars have even in times not so remote
by being accused of heresy and witchcraft. As late as
1522, in a biography published at Venice, it is related
that by black art he attained the papacy, after having
given his soul to the devil.2 Gerbert was, however,
interested in astrology,3 although this, was merely the
astronomy of that time and was such a science as any
learned man would wish to know, even as to-day we wish
to be reasonably familiar with physics and chemistry.

That Gerbert and his pupils knew the gobar numer-
als is a fact no longer open to controversy.4 Berneli-
nus and Richer6 call them by the well-known name of

1 Picavet, loc. cit., p. 37.

2 “ Con sinistre arti conseguri la dignita del Pontificato. . . . La-
sciato poi P abito, e’l monasterio, e datosi tutto in potere del diavolo.”
[Quoted in Bombelli, L'antica numerazione Italica , Rome, 1876, p. 41 n.]

8 He writes from Rheims in 984 to one Lupitus, in Barcelona, say-
ing: “ Itaque librum de astrologia translatum a te michi petenti di-
rige,” presumably referring to some Arabic treatise. [Epist. no. 24
of the Havet collection, p. 19.] 4 See Bubnov, loc. cit., p. x.

5 Olleris, loc. cit., p. 361, 1. 15, for Bernelinus ; and Bubnov, loc. cit.,
p. 381, 1. 4, for Richer.

DEFINITE INTRODUCTION INTO EUROPE 113

“ caracteres,” a word used by Radulph of Laon in the
same sense a century later.1 It is probable that Gerbert
was the first to describe these gobar numerals in any
scientific way in Christian Europe, but without the zero.
If he knew the latter he certainly did not understand
its use.2

The question still to be settled is as to where he
found these numerals. That he did not bring them from
Spain is the opinion of a number of careful investiga-
tors.3 This is thought to be the more probable because
most of the men who made Spain famous for learning
lived after Gerbert was there. Such were Ibn Slna
(Avicenna) who lived at the beginning, and Gerber of
Seville who flourished in the middle, of the eleventh
century, and Abu Roslid (Averroes) who lived at the
end of the twelfth.4 Others hold that his proximity to

1 Woepcke found this in a Paris MS. of Radulph of Laon, c. 1100.
[Propagation, p. 246.] “ Et prima quidem trium spaciorum superductio
unitatis caractere inscribitur, qui chakleo nomine dicitur igin .” See
also Alfred Nagl, “ Der arithmetische Tractat des Radulph von
Laon” ( Abhandlungen zur Geschichte der Mathematik , Vol.Y, pp. 85-
133), p. 97.

2 Weissenborn, loc. cit., p. 239. When Olleris ( CEuvres de Gerbert ,
Paris, 1867, p. cci) says, “ C’est h lui et non point aux Arabes, que
l’Europe doit son systfeme et sessignes de numeration,” he exaggerates,
since the evidence is all against his knowing the place value. Friedlein
emphasizes this in the Zeitschrift fur Mathematik und Physik, Vol. XII
(1867), Literaturzeitung , p. 70: “ Fur das System unserer Numeration
ist die Nidi das wesentlichste Merkmal, und diese kannte Gerbert nicht.
Er selbst schrieb alle Zahlen mit den romischen Zahlzeichen und man
kann ilnn also nicht verdanken, was er selbst nicht kannte.”

8 E.g., Chasles, Budinger, Gerliardt, and Richer. So Martin (Re-
cherches nouvelles etc.) believes that Gerbert received them from Boe-
thius or his followers. See Woepcke, Propagation , p. 41.

4 Budinger, loc. cit., p. 10. Nevertheless, in Gcrbert’s time one Al-
Mansur, governing Spain under the name of Hisham (976-1002), called
from the Orient Al-Begani to teach his son, so that scholars were
recognized. [Picavet, p. 30.]

114

THE HINDU- ARABIC NUMERALS

the Arabs for three years makes it probable that he as-
similated some of their learning, in spite of the fact
that the lines between Christian and Moor at that time
were sharply drawn.1 Writers fail, however, to recog-
nize that a commercial numeral system would have
been more likely to be made known by merchants than
by scholars. The itinerant peddler knew no forbidden
pale in Spain, any more than he has known one in other
lands. If the gobar numerals were used for marking
wares or keeping simple accounts, it was he who would
have known them, and who would have been the one
rather than any Arab scholar to bring them to the in-
quiring mind of the young French monk. The facts
that Gerbert knew them only imperfectly, that he used
them solely for calculations, and that the forms are evi-
dently like the Spanish gobar, make it all the more
probable that it was through the small tradesman of the
Moors that this versatile scholar derived his knowledge.
Moreover the part of the geometry bearing his name, and
that seems unquestionably his, shows the Arab influence,
proving that he at least came into contact with the
transplanted Oriental learning, even though imperfectly.2
There was also the persistent Jewish merchant trading
with both peoples then as now, always alive to the ac-
quiring of useful knowledge, and it would be very natu-
ral for a man like Gerbert to welcome learning from
such a source.

On the other hand, the two leading sources of infor-
mation as to the life of Gerbert reveal practically noth-
ing to show that he came within the Moorish sphere of
influence during his sojourn in Spain. These sources

1 Weissenborn, loc. cit. , p. 235. 2 Ibid., p. 234.

DEFINITE INTRODUCTION INTO EUROPE 115

are his letters and the history written by Richer. Gerbert
was a master of the epistolary art, and his exalted posi-
tion led to the preservation of his letters to a degree
that would not have been vouchsafed even by their
classic excellence.1 Richer was a monk at St. Remi de
Rheims, and was doubtless a pupil of Gerbert. The lat-
ter, when archbishop of Rheims, asked Richer to write a
history of his times, and this was done. The work lay
in manuscript, entirely forgotten until Pertz discovered
it at Bamberg in 1833.2 The work is dedicated to Ger-
bert as archbishop of Rheims,3 and would assuredly have
testified to such efforts as he may have made to secure
the learning of the Moors.

Now it is a fact that neither the letters nor this his-
tory makes any statement as to Gerbert’s contact with
the Saracens. The letters do not speak of the Moors,
of the Arab numerals, nor of Cordova. Spam is not
referred to by that name, and only one Spanish scholar
is mentioned. In one of his letters he speaks of Joseph
Ispanus,4 or Joseph Sapiens, but who this Joseph the
Wise of Spam may have been we do not know. Possibly

1 These letters, of the period 983-997, were edited by Havet, loc.
cit., and, less completely, by Olleris, loc. cit. Those touching mathe-
matical topics were edited by Bubnov, loc. cit., pp. 98-106.

2 He published it in the Monumenta Germaniae historica, “ Scrip-
tores,” Vol. Ill, and at least three other editions have since ap-
peared, viz. those by Guadet in 1845, by Poinsignon in 1856, and by
Waitz in 1877.

8 Domino ac beatissimo Patri Gerberto, Remorum archiepiscopo,
Riclierus Monchus, Gallorum congressibus in volumine regerendis,
imperii tui, pater sanctissime Gerberte, auctorjtas seminarium dedit.

4 In epistle 17 (Ilavet collection) he speaks of the “ De multipli-
catione et divisione numerorum libellum a Joseph Ispano editum abbas
Warnerius” (a person otherwise unknown). In epistle 25 he says:
“De multiplicatione et divisione numerorum, Joseph Sapiens sen-
tentias quasdam edidit.”

116

THE HINHU-ARABIC NUMERALS

it was lie who contributed the morsel of knowledge so
imperfectly assimilated by the young French monk.1
Within a few years after Gerbert’s visit two young Span-
ish monks of lesser fame, and doubtless with not that
keen interest in mathematical matters which Gerbert had,
regarded the apparently slight knowledge which they had
of the Hindu numeral forms as worthy of somewhat per-
manent record2 in manuscripts which they were transcrib-
ing. The fact that such knowledge had penetrated to their
modest cloisters in northern Spam — the one Albelda or
Albaida — indicates that it was rather widely diffused.

Gerbert’s treatise Libellus de numerorum divisione 3 is
characterized by Chasles as “ one of the most obscure
documents in the history of science.” 4 The most com-
plete information in regard to this and the other mathe-
matical works of Gerbert is given by Bubnov,5 who
considers this work to be genuine.6

1 H. Suter, “ Zur Frage fiber den Josephus Sapiens,” Bibliotheca
Mathematical Vol. VIII (2), p. 84; Weissenborn, Einfiihrung, p. 14;
also his Gerbert-, M. Steinschneider, in Bibliotheca Mathematica, 1893,
p. 68. Wallis ( Algebra , 1685, chap. 14) went over the list of Spanish
Josephs very carefully, but could find nothing save that “Josephus
Hispanus seu Josephus sapiens videtur aut Maurus fuisse aut alius
quis in Hispania.”

2 P. Ewald, Mittheilungen , Neues Archiv d. Gesellschaft fur dltere
deutsche Geschichtskunde , Vol. VIII, 1883, pp. 354-364. One of the
manuscripts is of 976 a. d. and the other of 992 a.d. See also Franz
Steffens, Lateinische Palaographie , Freiburg (Schweiz), 1903, pp.
xxxix-xl. The forms are reproduced in the plate on page 140.

8 It is entitled Constantino suo Gerbertus scolasticus , because it was
addressed to Constantine, a monk of the Abbey of Fleury. The text
of the letter to Constantine, preceding the treatise on the Abacus, is
given in the Comptes rendus , Vol. XVI (1843), p. 295. This book seems
to have been written c. 980 a.d. [Bubnov, loc. cit., p. 6.]

4 “ Ilistoire de l’Arithnkitique,” Comptes rendus , Vol. XVI (1843),
pp. 156, 281. 6 Loc. cit., Gerberti Opera etc.

0 Friedlein thought it spurious. See Zeitschrift fiir Mathematik und
Pliysik , Vol. XII (1867), Hist. -lit. suppl., p. 74. It was discovered in

DEFINITE INTRODUCTION INTO EUROPE 117

So little did Gerbert appreciate these numerals that
in his works known as the liegula de abaco computi and
the Libellus he makes no use of them at all, employing
only the Roman forms.1 Nevertheless Bernelinus 2 refers
to the nine gobar characters.3 These Gerbert had marked
on a thousand jetons or counters,4 using the latter on an
abacus which he had a sign-maker prepare for him.6
Instead of putting eight counters in say the tens’ column,
Gerbert would put a single counter marked 8, and so
for the other places, leaving the column empty where
we would place a zero, but where he, lacking the zero,
had no counter to place. These counters he possibly
called caracteres, a name which adhered also to the fig-
ures themselves. It is an interesting speculation to con-
sider whether these apices , as they are called in the
Boethius interpolations, were in any way suggested by
those Roman jetons generally known in numismatics
as tesserae, and bearing the figures I-XVI, the sixteen
referring to the number of assi in a sestertius .6 The

the library of the Benedictine monastry of St. Peter, at Salzburg, and
was published by Peter Bernhard Pez in 1721. Doubt was first cast
upon, it in the Olleris edition ( CEuvres de Gerbert). See Weissenborn,
Gerbert , pp. 2, 6, 168, and Picavet, p. 81. Hock, Cantor, and Tli. Martin
place the composition of the work at c. 996 when Gerbert was in Ger-
many, while Olleris and Picavet refer it to the period when he was at
Rheims.

1 Picavet, loc. cit., p. 182.

2 Who wrote after Gerbert became pope, for he uses, in his preface,
the words, “a domino pape Gerberto.” He was quite certainly not
later than the eleventh century; we do not have exact information
about the time in which he lived.

8 Picavet, loc. cit., p. 182. Weissenborn, Gerbert , p. 227. In Olleris,
Liber Abaci (of Bernelinus), p. 361.

4 Richer, in Bubnov, loc. cit., p. 381.

5 Weissenborn, Gerbert , p. 241.

6 Writers on numismatics are quite uncertain as to their use. See
F. Gnecchi, Monete Komane , 2d ed., Milan, 1900, cap. XXXVII. For

118

THE HINDU-ARABIC NUMERALS

name apices adhered to the Hindu-Arabic numerals until
the sixteenth century.1

To the figures on the apices were given the names
Igin, andras, ormis, arbas, quimas, calctis or caltis, zenis,
temenias, celentis, sipos,2 the origin and meaning of
which still remain a mystery. The Semitic origin of
several of the words seems probable. Wahud , thaneine,

pictures of old Greek tesserae of Sarmatia, see S. Ambrosoli, Monete
Greche, Milan, 1899, p. 202.

1 Thus Tzwivel’s arithmetic of 1507, fol. 2, v., speaks of the ten fig-
ures as “ cliaracteres sive numerorum apices a diuo Seuerino Boetio.”

2 Weissenborn uses sipos for 0. It is not given by Bernelinus, and
appears in Radulph of Laon, in the twelfth century. See Gunther’s
Geschichte, p. 98, n. ; Weissenborn, p. 11 ; Pilian, Expos6 etc., pp.
xvi-xxii.

In Friedlein’s Boetius , p. 39G, the plate shows that all of the six im-
portant manuscripts from which the illustrations are taken contain the
symbol, while four out of five which give the words use the word sipos
for 0. The names appear in a twelfth-century anonymous manuscript
in the Vatican, in a passage beginning

Ordine primigeno sibi nomen possidet igin.

Andras ecce locum mox uendicat ipse secundum

Ormis post numeros incompositus sibi primus.

[Boncompagni Bulletino, XV, p. 132.] Turchill (twelfth century) gives
the names Igin, andras, hormis, arbas, quimas, caletis, zenis, temenias,
celentis, saying : “ Has autem figuras, ut donnus [dominus] Gvillelmus
Rx testatur, a pytagoricis habemus, nomina uero ab arabibus.” (Who
the William R. was is not known. Boncompagni Bulletino XV, p. 136.)
Radulph of Laon (d. 1131) asserted that they were Chaldean ( Propa-
gation, p. 48 n.). A discussion of the whole question is also given in
E. C. Bayley, loc. cit. Huet, writing in 1679, asserted that they were
of Semitic origin, as did Nesselmann in spite of his despair over ormis,
calctis, and celentis; see Woepcke, Propagation , p. 48. The names
were used as late as the fifteenth century, without the zero, but with
the superscript dot for 10’s, two dots for 100’s, etc., as among the
early Arabs. Gerliardt mentions having seen a fourteenth or fifteenth
century manuscript in the Bibliotheca Amploniana with the names
“ Ingnin, andras, armis, arbas, quinas, calctis, zencis, zemenias, zcelen-
tis,” and the statement “ Si unum punctum super ingnin ponitur, X
significat. ... Si duo puncta super . . . figuras superponunter, fiet
decuplim illius quod cum uno puncto significabatur,” in Monats-
berichtc der K. P. Akad. d. lUiss., Berlin, 1867, p. 40.

DEFINITE INTRODUCTION INTO EUROPE 119

thalata, arba, Jcumsa, sett a, sebba , timinia , taseucl are given
by the Rev. R. Patrick 1 as the names, in an Arabic dia-
lect used hi Morocco, for the numerals from one to nine.
Of these the words for four, five, and eight are strikingly
like those given above.

The name apices was not, however, a common one in
later times. Notae was more often used, and it finally
gave the name to notation.2 Still more common were
the names figures , ciphers , signs , elements, and characters .3

So little effect did the teachings of Gerbert have in
making known the new numerals, that O’Creat, who
lived a century later, a friend and pupil of Adelhard

1 A chart of ten numerals in 200 tongues , by Rev. R. Patrick, Lon-
don, 1812.

2 “ Nuirieratio figuralis est cuiusuis numeri per notas, et figuras
numerates descriptio.” [Clicktoveus, edition of c. 1507, fol. C ii, v.]
“Aristoteles enim uoces rerum t vjg.fio\a uocat: id translatum, sonat
notas.” [Noviomagus, De Numeris Lilnri II, cap. vi.] “Alphabetum
decern notarum.” [Schonerus, notes to Ramus, 1586, p. 3 seq.] Richer
says: “ novem numero notas omnem numerum significantes.” [Bubnov,
loc. cit., p. 381.]

3 “ II y a dix Characteres, autrement Figures, Notes, ou Elements.”
[Peletier, edition of 1607, p. 13.] “ Numerorum notas alij figuras, alij
signa, alij characteres uocant.” [Glareanus, 1545 edition, f. 9, r.j
“ Per figuras (quas zyphras uocant) assignationem, quales sunt lise
notulse, 1. 2. 3. 4. . . .” [Noviomagus, De Numeris Libri II, cap. vi.]
Gemma Frisius also uses elementa and Cardan uses literae. In the first
arithmetic by an American (Greenwood, 1729) the author speaks of
“a few Arabian Charecters or Numeral Figures, called Digits ” (p. 1),
and as late as 1790, in the third edition of J. J. Blassifere’s arithmetic
(1st ed. 1769), the name characters is still in use, both for “ de Latynsche
en de Arabische ” (p. 4), as is also the term “ Cyfferletters ” (p. 6, n.).
Ziffer, the modern German form of cipher, was commonly used to
designate any of the nine figures, as by Boeschenstein and Riese,
although others, like Kobel, used it only for the zero. So zifre ap-
pears in the arithmetic by Borgo, 1550 ed. In a Munich codex of the
twelfth century, attributed to Gerland, they are called characters only :
“ Usque ad VIIII. enim porrigitur omnis numerus et qui supercrescit
eisdem designator Karacteribus.” [Boncompagni Bulletino, Vol. X,
p. 607.]

120

the iiindu-arabic numerals

of Bath, used the zero with the Roman characters, in
contrast to Gerbert’s use of the gobar forms without
the zero.1 O’Creat uses three forms for zero, o, 5, and
t, as in Maximus Planudes. With this use of the zero
goes, naturally, a place value, for he writes III III for
33, ICCOO and I. II. t. t for 1200, 1. 0. VIII. IX for 1089,
and I. IIII. IIII. tttt for the square of 1200.

The period from the time of Gerbert until after the
appearance of Leonardo’s monumental work may be
called the period of the abacists. Even for many years
after the appearance early in the twelfth century of the
books explaining the Hindu art of reckoning, there was
strife between the abacists, the advocates of the abacus,
and the algorists, those who favored the new numerals.
The words cifra and algorismus cifra were used with
a somewhat derisive significance, indicative of absolute
uselessness, as indeed the zero is useless on an abacus
in which the value of any unit is given by the column
which it occupies.2 So Gautier de Coincy (1177-1236)
in a work on the miracles of Mary says :

A horned beast, a sheep,

An algorismus-cipher,

Is a priest, who on such a feast day
Does not celebrate the holy Mother.3

So the abacus held the field for a long time, even
against the new algorism employing the new numerals.

1 The title of his work is Prologue N. Ocreati in Helceph (Arabic
al-qeif , investigation or memoir) ad Adelardum Batensem magistrum
suum. The work was made known by C. Henry, in the Zeitschrift fur
Mathematik und Physik , Vol. XXV, p. 129, and in the Abhandlungen
zur GeschicMe der Mathematik , Vol. Ill ; Weissenborn, Gerbert , p. 188.

2 The zero is indicated by a vacant column.

8 Leo Jordan, loc. cit., p. 170. “Cliifre en augorisme” is the ex-
pression used, while a century later “giffre en argorisme ” and “ cyffres
d’augorisme” are similarly used.

DEFINITE INTRODUCTION INTO EUROPE 121

Geoffrey Chaucer 1 describes in The Miller's Tale the clerk

with « pi is Almageste and bokcs grete and smale,

Ills astrelabie, longinge for his art,

Ilis angrim-stones layen faire apart
On shelves couched at his beddes heed.”

So, too, in Chaucer’s explanation of the astrolabe,2
written for his son Lewis, the number of degrees is ex-
pressed on the instrument in Hindu -Arabic numerals :
“ Over the whiche degrees ther ben noumbres of augrim,
that devyden thilke same degrees fro fyve to fyve,”
and “ . . . the nombres . . . ben writen hi augrim,”
meaning in the way of the algorism. Thomas Usk
about 1887 writes : 3 “a sypher in augrim have no might
in signification of it-selve, yet he yeveth power in sig-
nification to other.” So slow and so painful is the assimi-
lation of new ideas.

Bernelinus4 states that the abacus is a well-polished
board (or table), which is covered with blue sand and
used by geometers in drawing geometrical figures. We
have previously mentioned the fact that the Hindus also
performed mathematical computations in the sand, al-
though there is no evidence to show that they had any
column abacus.6 For the purposes of computation,
Bernelinus continues, the board is divided into thirty
vertical columns, three of which are reserved for frac-
tions. Beginning with the units columns, each set of

1 The Works of Geoffrey Chaucer , edited by W. W. Skeat, Vol. IV,
Oxford, 1894, p. 92.

2 Loc. cit., Vol. Ill, pp. 179 and 180.

8 In Book II, chap, vii, of The Testament of Love , printed with
Chaucer’s Works, loc. cit., Vol. VII, London, 1897.

4 Liber Abacci , published in Olleris, CEuvres de Gerbert , pp. 357-400.

6 G. R. Kaye, “The Use of the Abacus in Ancient India,” Journal
and Proceedings of the Asiatic Society of Bengal, 1908, pp. 293-297.

122

THE IIINDU-ARABIC NUMERALS

three columns ( lineae is the word which Bemelinus uses)
is grouped together by a semicircular arc placed above
them, while a smaller arc is placed over the units col-
umn and another joins the tens and hundreds columns.
Thus arose the designation arcus pictagore 1 or sometimes
simply arcus.1 2 The operations of addition, subtraction,
and multiplication upon this form of the abacus required
little explanation, although they were rather extensively
treated, especially the multiplication of different orders
of numbers. But the operation of division was effected
with some difficulty. F or the explanation of the method
of division by the use of the complementary difference,3
long the stumbling-block in the way of the medieval
arithmetician, the reader is referred to works on the his-
tory of mathematics4 and to works relating particularly
to the abacus.5

Among the writers on the subject may be mentioned
Abbo6 of Fleury (c. 970), Heriger 7 of Lobbes or Laubacli

1 Liber Abbaci , by Leonardo Pisano, loc. cit., p. 1.

2 Friedlein, “ Die Entwickelung des Rechnensmit Columnen,” Zeit-
schriftfur Mathematik und Physik, Vol. X, p. 247.

3 The divisor 6 or 16 being increased by the difference 4, to 10 or
20 respectively.

4 E.g. Cantor, Vol. I, p. 882.

6 Friedlein, loc. cit.; Friedlein, “Gerbert’s Regeln der Division”
and “Das Reclinen mit Columnen vor deni 10. Jalirhundert,” Zeit-
schrift fur Mathematik und Physik , Vol. IX ; Bubnov, loc. cit., pp. 197-
245; M. Chasles, “Ilistoire de l’arithm&ique. Recherclies des traces
du syst6ine de l’abacus, aprfes que cette nidthode a pris le nom d’Algo-
risme. — Preuves qu’& toutes les ^poques, jusqu’au xvic sifecle, on a su
que l’arithmdtique vulgaire avait pour origine cette m<5tliode ancienne,”
Comptes rendus , Vol. XVII, pp. 143-154, also “ Regies de l'abacus,”
Comptes rendus , Vol. XVI, pp. 218-246, and “Analyse et explication
du traits de Gerbert,” Comptes rendus, Vol. XVI, pp. 281-299.

6 Bubnov, loc. cit., pp. 203-204, “ Abbonis abacus.”

7 “Regulae de numerorum abaci rationibus,” in Bubnov, loc. cit.,
pp. 205-225.

DEFINITE INTRODUCTION INTO EUROPE 123

(c. 950-1007), and Hermannus Contractus1 (1013-
1054), all of whom employed only the lloman numerals.
Similarly Adelhard of Bath (c. 1130), in his work lieyulae
Abaci ,2 gives no reference to the new numerals, although it
is certain that he knew them. Other writers on the abacus
who used some form of Hindu numerals were Gerland 3
(first half of twelfth century) and Turchill4 (c. 1200).
For the forms used at this period the reader is referred
to the plate on page 88.

After Gerbert’s death, little by little the scholars of
Europe came to know the new figures, chiefly through
the introduction of Arab learning. The Dark Ages had
passed, although arithmetic did not find another advo-
cate as prominent as Gerbert for two centuries. Speak-
ing of this great revival, Raoul Glaber5 6 * (985~c. 1046), a
monk of the great Benedictine abbey of Cluny, of the
eleventh century, says : “ It was as though the world had
arisen and tossed aside the worn-out garments of ancient
time, and wished to apparel itself in a white robe of
churches.” And with this activity in religion came a
corresponding interest in other lures. Algorisms began
to appear, and knowledge from the outside world found

1 P. Treutlein, “Intorno ad alcuni scritti inediti relativi al calcolo
dell1 abaco,11 Bulletino di bibliografia e di storia delle scienze matema-
ticlie efisiche, Vol. X, pp. 589-647.

2 “Intorno ad uno scritto inedito di Adelhardo di Bath intitolato
‘Regulae Abaci,1 11 B. Boncompagni, in his Bulletino, Vol. XIV,
pp. 1-134.

8 Treutlein, loc. cit. ; Boncompagni, “ Intorno al Tractatus de Abaco
di Gerlando,” Bulletino , Vol. X, pp. G48-G5G.

4 E. Narducci, “Intorno a due trattati inediti d’abaco contenuti

in due codici Vaticani del secolo XII,” Boncompagni Bulletino, Vol.
XV, pp. 111-162.

6 See Molinier, Les sources de Vhistoire de France, Vol. II, Paris,

1902, pp. 2, 3.

124

THE IIINDU-ARABTC NUMERALS

interested listeners. Another Raoul, or Radulph, to whom
we have referred as Radulph of Laon,1 a teacher in the
cloister school of his city, and the brother of Anselm of
Laon 2 the celebrated theologian, wrote a treatise on music,
extant but unpublished, and an arithmetic which Nagl
first published in 1890. 3 The latter work, preserved to us
hi a parchment manuscript of seventy-seven leaves, con-
tains a curious mixture of Roman and gobar numerals, the
former for expressing large results, the latter for practical
calculation. These gobar “ caracteres ” include the sipos
(zero), O, of which, however, Radulph did not know
the full significance; showing that at the opening of the
twelfth century the system was still uncertain in its status
in the church schools of central France.

At the same time the words algorismus and cifra were
coming into general use even in non-mathematical litera-
ture. J ordan 4 cites numerous instances of such use from
the works of Alanus ab Insulis 6 (Alain cle Lille), Gau-
tier cle Coincy (1177-1236), and others.

Another contributor to arithmetic during this interest-
ing period was a prominent Spanish J ew called variously
John of Luna, John of Seville, Johannes Hispalensis,
Johannes Toletanus, and Johannes Ilispanensis de Luna.6

1 Cantor, Geschichte, Vol. I, p. 702. A. Nagl in the Abhandlungen
zur Geschichte der Mathematik , Vol. V, p. 85.

2 1030-1117.

8 Abhandlungen zur Geschichte der Mathematik , Vol. V, pp. 85-133.
The work begins “ Incipit Liber Radulfi laudunensis de abaco.”

4 Materialien zur Geschichte der arabischen Zahlzeichen in Frankreich,
loc. cit. 5 Who died in 1202.

8 Cantor, Geschichte, Vol. I (3), pp. 800-803 ; Boncompagni, Trattati ,
Part II. M. Steinschneider (“Die Mathematik bei den Juden,”
Bibliotheca Mathematica, Vol. X (2), p. 70) ingeniously derives another
name by which he is called (Abendeuth) from Ibn Daud (Son of David).
See also Abhandlungen , Vol. Ill, p. 110.

DEFINITE INTRODUCTION INTO EUROPE 125

His date is rather closely fixed by the fact that he dedi-
cated a work to Raimund who was archbishop of Toledo
between 1130 and 1150.1 Ilis interests were chiefly in
the translation of Arabic works, especially such as bore
upon the Aristotelian philosophy. From the standpoint
of arithmetic, however, the chief interest centers about a
manuscript entitled Joannis Hispalensis liber Algorismi de
Practica Ariametrice which Boncompagni found in what
is now the Bibliotheque nationale at Paris. Although this
distinctly lays claim to being Al-Ivhowarazmi’s work,2
the evidence is altogether against the statement,3 hut the
hook is quite as valuable, since it represents the knowl-
edge of the time in which it was written. It relates to the
operations with integers and sexagesimal fractions, in-
cluding roots, and contains no applications.4

Contemporary with John of Luna, and also living in
Toledo, was Gherard of Cremona,5 who has sometimes
been identified, but erroneously, with Gernardus,6 the

1 John is said to have died in 1157.

2 For it says, “Incipit prologus in libro alghoarismi de practica
arismetrice. Qui editus est a magistro Johanne yspalensi.” It is pub-
lished in full in the second part of Boncompagni’s Traltati d'aritmctica.

3 Possibly, indeed, the meaning of “libro alghoarismi” is not “to
Al-Khowarazmi’s book,” but “to a book of algorism.” John of Luna
says of it: “Hoc idem est illud etiam quod . . . alcorismus dicere
videtur.” [Traltati, p. 68.]

4 For a rfeum^, see Cantor, Vol. I (3), pp. 800-803. As to the au-

thor, see Enestrom in the Bibliotheca Mathematica, Vol. VI (3), p. 114,
and Vol. IX (3), p. 2.

6 Born at Cremona (although some have asserted at Carmona, in
Andalusia) in 1114; died at Toledo in 1187. Cantor, loc. cit.; Bon-
compagni, Atti d. It. Accad. d. n. Lined, 1851.

6 See Abhandlungen zur Geschichte der Mathematik, Vol. XIV, p. 149 ;
Bibliotheca Mathematica, Vol. IV (3), p. 206. Boncompagni had a
fourteenth-century manuscript of his work, Gerardi Cremonensis artis
metrice practice. See also T. L. Heath, The Thirteen Books of Euclid's
Elements , 3 vols., Cambridge, 1908, Vol. I, pp. 92-94 ; A. A. Bjornbo,

126

THE IilNDU-ARABIC NUMERALS

author of a work on algorism. He was a physician, an
astronomer, and a mathematician, translating from the
Arabic both in Italy and in Spain. In arithmetic he was
influential in spreading the ideas of algorism.

Four Englishmen — Adelhard of Bath (c. 1130), Rob-
ert of Chester (Robertas Cestrensis, c. 1143), William
Shelley, and Daniel Morley (1180) — are known 1 to
have journeyed to Spain in the twelfth century for the
purpose of studying mathematics and Arabic. Adelhard
of Bath made translations from Arabic into Latin of Al-
Kliowarazmfs astronomical tables2 and of Euclid’s Ele-
ments,3 while Robert of Chester is known as the translator
of Al-Ivhowarazmi’s algebra.4 There is no reason to doubt
that all of these men, and others, were familiar with the
numerals which the Arabs were using.

The earliest trace we have of computation with Hindu
numerals in Germany is in an Algorismus of 1143, now
in the Hofbibliothek in Vienna.6 It is bound in with a

“Gerhard von Cremonas Ubersetzung von Alkwarizmis Algebra und
von Euklids Elementen,” Bibliotheca Mathematical Yol. VI (3), pp.
239-248. 1 Wallis, Algebra , 1685, p. 12 seq.

2 Cantor, Geschichte, \ ol. 1(3), p. 906; A. A. Bjornbo, “Al-Cliwa-
rizmi’s trigononietriske Xavier,” Festskrift til II. G. Zeuthen , Copen-
hagen, 1909, pp. 1-17. 3 Heath, loc. cit., pp. 93-96.

4 M. Steinschneider, Zeitschrift der deutschen morgenlandischen Ge-
sellschaft, Vol. XXV, 1871, p. 104, and Zeitschrift fur Mathematik und
Fhysilc, Vol. XVI, 1871, pp. 392-393; M. Curtze, Centralblatt fur
Bibliothekswesen, 1899, p. 289 ; E. Wappler, Zur Geschichte der deut-
schen Algebra im 15. Jahrhundert, Progrannn, Zwickau, 1887 ; L. C.
Karpinski, “Robert of Chester’s Translation of the Algebra of Al-
Khowarazml,” Bibliotheca Mathematical Vol. XI (3), p. 125. He is also
known as Robertus Retinensis, or Robert of Reading.

6 Nagl, A., “Ueher eine Algorismus- Schrift des XII. Jalirhunderts
und liber die Verbreitung der indiscli-arabischen Reclienkunst und
Zalilzeichen im christl. Abendlande,” in the Zeitschrift fur Mathematik
und Physik , Ilist. -lit. Abth., Vol. XXXIV, p. 129. Curtze, Abhand-
lungen zur Geschichte der Mathematik , Vol, VIII, pp, 1-27,

DEFINITE INTRODUCTION INTO EUROPE 127

Computus by the same author and bearing the date given.
It contains chapters “ De additione,” “ De diminutione,”
“ De mediatione,” “ De divisione,” and part of a chap-
ter on multiplication. The numerals are in the usual medi-
eval forms except the 2, which, as will be seen from the
illustration,1 is somewhat different, and the 3, which
takes the peculiar shape h, a form characteristic of the
twelfth century.

It was about the same time that the Sefer ha-Mispar ,2
the Book of Number, appeared in the Hebrew language.
The author, Rabbi Abraham ibn Me'ir ibn Ezra,3 was
bom in Toledo (c. 1092). In 1139 he went to Egypt,
Palestine, and the Orient, spending also some years in
Italy. Later he lived in southern France and in Eng-
land. He died in 1167. The probability is that he ac-
quired his knowledge of the Hindu arithmetic4 in his
native town of Toledo, but it is also likely that the
knowledge of other systems which he acquired on travels
increased his appreciation of this one. We have men-
tioned the fact that he used the first letters of the Hebrew
alphabet, to n 1 1 n 1 3D N, for the

numerals 9 8 7 G 5 4 32 1, and a

circle for the zero. The quotation in the note given be-
low shows that he knew of the Hindu origin ; but in his
manuscript, although he set down the Hindu forms, he
used the above nine Hebrew letters with place value for
all computations.

1 See line a in the plate on p. 143.

2 Sefer ha-Mispar , Das Buck der Zahl, ein hebraisch-arithmetisches
Werlc des R. Abrahamibn Esra , Moritz Silberberg, Frankfurt a.M., 1895.

3 Browning’s “ Rabbi ben Ezra.”

4 “Darum haben auch die Weisen Indiens all ilire Zalilen durch
neun bezeiclinet und Formen fur die 9 Ziffern gebildet.” [Sefer ha-
Mispar, loc. cit., p. 2.1

CHAPTER VIII

THE SPREAD OF THE NUMERALS IN EUROPE

Of all the medieval writers, probably the one most in-
fluential in introducing the new numerals to the scholars
of Europe was Leonardo Fibonacci, of Pisa.1 This remark-
able man, the most noteworthy mathematical genius of
the Middle Ages, was born at Pisa about 1175.2

The traveler of to-day may cross the Via Fibonacci
on bis way to the Campo Santo, and there be may see
at the end of the long corridor, across the quadrangle,
the statue of Leonardo in scholar’s garb. Few towns
have honored a mathematician more, and few mathema-
ticians have so distinctly honored their birthplace. Leo-
nardo was born hi the golden age of this city, the period
of its commercial, religious, and intellectual prosperity.3

1 F. Bonaini, “Memoria unica sincrona di Leonardo Fibonacci,”
Pisa, 1858, republished in 1867, and appearing in the Giornale Arca-
dico, Vol. CXCVII (N. S. LII); Gaetano Milanesi, Documento inedito e
sconosciuto a Lionardo Fibonacci, Roma, 1867 ; Guglielmini, Elogio
di Lionardo Pisano , Bologna, 1812, p. 35; Libri, Histoire des sci-
ences matMmatiques , Yol. II, p. 25; D. Martines, Origine e progressi
dell' aritmetica , Messina, 1865, p. 47 ; Lucas, in Boncompagni Bulle-
tino , Yol. X, pp. 129, 239 ; Besagne, ibid., Vol. IX, p. 583; Boncompagni,
three works as cited in Chap. I; G. Enestrom, “Ueber zwei angeb-
liclie mathematische Schulen im christliclien Mittelalter,” Bibliotheca
Mathematical Vol. VIII (3), pp. 252-262; Boncompagni, “Della vita
e delle opere di Leonardo Pisano,” loc. cit.

2 The date is purely conjectural. See the Bibliotheca Mathematica ,
Vol. IV (3), p. 215.

8 An old chronicle relates that in 1063 Pisa fought a great battle
with the Saracens at Palermo, capturing six ships, one being “full of
wondrous treasure,” and this was devoted to building the cathedral.

128

SPREAD OF THE NUMERALS IN EUROPE 129

Situated practically at the mouth of the Arno, Pisa
formed with Genoa and Venice the trio of the greatest
commercial centers of Italy at the opening of the thirteenth
century. Even before Venice had captured the Levan-
tine trade, Pisa had close relations with the East. An
old Latin chronicle relates that in 1005 “Pisa was cap-
tured by the Saracens,” that in the following year “ the
Pisans overthrew the Saracens at Reggio,” and that in
1012 “ the Saracens came to Pisa and destroyed it.” The
city soon recovered, however, sending no fewer than a
hundred and twenty ships to Syria in 1099, 1 founding
a merchant colony in Constantinople a few years later,2
and meanwhile carrying on an mterurban warfare in Italy
that seemed to stimulate it to great activity.3 A writer
of 1114 tells us that at that time there were many hea-
then people — Turks, Libyans, Parthians, and Chalde-
ans— to be found in Pisa. It was in the midst of such
wars, in a cosmopolitan and commercial town, in a cen-
ter where literary work was not appreciated,4 that the
genius of Leonardo appears as one of the surprises of
history, warning us again that “we should draw no
horoscope ; that we should expect little, for what we
expect will not come to pass.” 5 *

Leonardo’s father was one William,0 and he had a
brother named Bonaccingus,7 but nothing further is

1 Heyd, loc. cit., Vol. I, p. 149. 2 Ibid., p. 211.

8 J. A. Symonds, Renaissance in Italy. The Age of Despots. New

York, 1883, p. 02. 4 Symonds, loc. cit., p. 79.

5 J. A. Froude, The Science of History , London, 1864. “Un brevet

d’apothicaire n’empecha pas Dante d’etre le plus grand pofete de
l’ltalie, et ce fut un petit marcliand de Fise qui donna l’algfebre aux

Chretiens.” [Libri, Histone, Vol. I, p. xvi.]

0 A document of 1226, found and published in 1858, reads: “Leo-
nardo bigollo quondam Guilielmi.” 7 “Bonaccingo germano suo,”

130

THE IIINDU-ARABIC NUMERALS

known of his family. As to Fibonacci, most writers 1 have
assumed that his father’s name was Bonaccio,2 whence
films Bonaccii , or Fibonacci. Others3 believe that the
name, even in the Latin form of filius Bonaccii as used
in Leonardo’s work, was simply a general one, like our
Johnson or Bronson (Brown’s son); and the only con-
temporary evidence that we have bears out this view.
As to the name Bigollo, used by Leonardo, some have
thought it a self -assumed one meaning blockhead, a term
that had been applied to him by the commercial world
or possibly by the university circle, and taken by him
that he might prove what a blockhead could do. Mila-
nesi,4 however, has shown that the word Bigollo (or
Pigollo) was used in Tuscany to mean a traveler, and
was naturally assumed by one who had studied, as Leo-
nardo had, in foreign lands.

Leonardo’s father was a commercial agent at Bugia,
the modern Bougie,5 the ancient Saldae on the coast of
Barbary,6 a royal capital under the Vandals and again,
a century before Leonardo, under the Beni Ffammad.
It had one of the best harbors on the coast, sheltered as
it is by Mt. Lalla Guraia,7 and at the close of the twelfth
century it was a center of African commerce. It was here
that Leonardo was taken as a child, and here he went to
school to a Moorish master. When he reached the years
of young manhood he started on a tour of the Medi-
terranean Sea, and visited Egypt, Syria, Greece, Sicily,
and Provence, meeting with scholars as well as with

1 E.g. Libri, Guglielmim, Tirabosclii. 2 Latin, Bonciccius.

8 Boncompagni and Milanesi. 4 Reprint, p. 5.

6 Whence the Erench name for candle.

6 Now part of Algiers.

7 E. Eeclus, Africa, New York, 1893, Vol. II, p. 253.

SPREAD OF THE NUMERALS IN EUROPE 131

merchants, and imbibing a knowledge of the various sys-
tems of numbers in use in the centers of trade. All these
systems, however, he says he counted almost as errors
compared with that of the Hindus.1 Returning to Pisa,
he wrote his Liber Abaci 2 hi 1202, rewriting it in 1228.3
In this work the numerals are explained and are used
in the usual computations of business. Such a treatise
was not destined to be popular, however, because it
was too advanced for the mercantile class, and too
novel for the conservative university circles. Indeed, at
this time mathematics had only slight place in the newly
established universities, as witness the oldest known stat-
ute of the Sorbonne at Paris, dated 1215, where the sub-
ject is referred to only in an incidental way.4 The period
was one of great commercial activity, and on this very

1 “ Sed hoc totum et algorisrrram atque arcus pictagore quasi erro-
rem computavi respectu modi indorum.” Woepcke, Propagation etc.,
regards this as referring to two different systems, but the expression
may very well mean algorism as performed upon the Pythagorean
arcs (or table).

2 “ Book of the Abacus,” this term then being used, and long after-
wards in Italy, to mean merely the arithmetic of computation.

8 “ Incipit liber Abaci a Leonardo filio Bonacci compositus anno
1202 et correctus ab eodem anno 1228.” Three MSS. of the thirteenth
century are known, viz. at Milan, at Siena, and in the Vatican library.
The work was first printed by Boncompagni in 1857.

4 I.e. in relation to the quadrivium. “Non legant in festivis diebus,
nisi Philosoplios et rhetoricas et quadrivalia et barbarismuin et ethi-
cam, si placet.” Suter, Die Mathematik auf den Universitaten des
MittelaUefrs, Zurich, 1887, p. 50. Roger Bacon gives a still more
gloomy view of Oxford in his time in his Ojms minus, in the Rerum
Britannicarum medii aevi scriptores, London, 1859, Vol. I, p. 327. For
a picture of Cambridge at this time consult F. W. Newman, The
English Universities, translated from the German of V. A. Huber, Lon-
don, 1843, Vol. I, p. 01; W. W. R. Ball, History of Mathematics at
Cambridge, 1889 ; S. Gunther, Geschiclite des mathematischen Unter-
riclits im deutschen Mittelalter bis zum Jahre 1525, Berlin, 1887, being
Vol. Ill of Monumenta Germaniae paedagogica,

132

THE IIINDU-ARABIC NUMERALS

account such a book would attract even less attention
than usual.1

It would now be thought that the western world
would at once adopt the new numerals which Leonardo
had made known, and which were so much superior to
anything that had been in use in Christian Europe. The
antagonism of the universities would avail but little, it
would seem, against such an improvement. It must be
remembered, however, that there was great difficulty in
spreading knowledge at this time, some two hundred and
fifty years before printing was invented. “Popes and
princes and even great religious institutions possessed far
fewer books than many farmers of the present age. The
library belonging to the Cathedral Church of San Mar-
tino at Lucca in the ninth century contained only nineteen
volumes of abridgments from ecclesiastical commenta-
ries.” 2 Indeed, it was not until the early part of the fif-
teenth century that Palla degli Strozzi took steps to carry
out the project that had been in the mind of Petrarch,
the founding of a public library. It was largely by
word of mouth, therefore, that this early knowledge had
to be transmitted. Fortunately the presence of foreign
students in Italy at this tune made this transmission
feasible. (If human nature was the same then as now, it
is not impossible that the very opposition of the faculties
to the works of Leonardo led the students to investigate

1 On the commercial activity of the period, it is known that bills
of exchange passed between Messina and Constantinople in 1161,
and that a bank was founded at Venice in 1170, the Bank of San
Marco being established in the following year, llie activity of 1 isa
was very manifest at this time. Heyd, loc. cit., Vol. II, p. 5 ; V. Casa-
grandi, Storia e cronologia, 3d ed., Milan, 1901, p. 56.

2 J. A. Symonds, loc. cit., Vol. II, p. 127.

SPREAD OF TIIE NUMERALS TN EUROPE 133

them the more zealously.) At Vicenza in 1209, for
example, there were Bohemians, Poles, Frenchmen,
Burgundians, Germans, and Spaniards, not to speak of
representatives of divers towns of Italy ; and what was
true there was also true of other intellectual cenfers.
The knowledge could not fail to spread, therefore, and
as a matter of fact we find numerous bits of evidence
that this was the case. Although the bankers of Flor-
ence were forbidden to use these numerals in 1299, and
the statutes of the university of Padua required station-
ers to keep the price lists of books “ non per cifras, sed
per literas claros,” 1 the numerals really made much
headway from about 1275 on.

It was, however, rather exceptional for the common
people of Germany to use the Arabic numerals before the
sixteenth century, a good witness to this fact being the
popular almanacs. Calendars of 1457-1496 2 have gener-
ally the Roman numerals, while Kobel’s calendar of 1518
gives the Arabic forms as subordinate to the Roman.
In the register of the Kreuzschule at Dresden the Roman
forms were used even until 1539.

While not minimizing the importance of the scientific
work of Leonardo of Pisa, we may note that the more pop-
ular treatises by Alexander de Villa Dei (c. 1240 a.d.)
and John of Halifax (Sacrobosco, c. 1250 a.d.) were
much more widely used, and doubtless contributed more
to the spread of the numerals among the common people.

1 I. Taylor, The Alphabet , London, 1883, Vol. II, p. 263.

2 Cited by Unger’s History, p. 15. The Arabic numerals appear in
a Regensburg chronicle of 1167 and in Silesia in 1340. See Schmidt’s
Encyclopadie der Erziehung, Vol. VI, p. 726 ; A. Kuckuk, “ Die Recken-
kunst im sechzehnten Jahrhundert,” Festschrift zur dritten Sacularfeier
des Berlinischen Gymnasiums zum grauen Kloster, Berlin, 1874, p. 4.

134

THE HINDU-ARABIC NUMERALS

The Carmen de Algorismo 1 of Alexander de Villa Dei
was written in verse, as indeed were many other text-
books of that time. That it was widely used is evidenced
by the large number of manuscripts 2 extant in European
libraries. Sacrobosco’s Algorismus ,3 hi which some lines
from the Carmen are quoted, enjoyed a wide popularity
as a textbook for university instruction.4 The work was
evidently written with this end in view, as numerous

commentaries by university lecturers are found. Proba-
bly the most widely used of these was that of Petrus de
Dacia5 written hi 1291. These works throw an interest-
ing light upon the method of instruction in mathematics
hi use hi the universities from the thirteenth even to the
sixteenth century. Evidently the text was first read and
copied by students.6 F ollowhig this came line by line an
exposition of the text, such as is given hi Petrus de
Dacia’s commentary.

Sacrobosco’s work is of hiterest also because it was
probably due to the extended use of this work that the

1 The text is given in Halliwell, Bara Mathematica, London, 1839.

2 Seven are given in Aslimole’s Catalogue of Manuscripts in the
Oxford Library , 1845.

8 Maximilian Curtze, Petri Philomeni de Dacia in Algorismum Vid-
garem Johannis de Sacrobosco commentarius , una cum Algorismo ipso ,
Copenhagen, 1897 ; L. C. Karpinski, “ Jordanus. Nemorarius and John
of Halifax,” American Mathematical Monthly, Vol. XVII, pp. 108-113.

4 J. Aschbach, Geschichte dcr Wiener Universitat im ersten Jahrhun-
derte Hires Bestehens, Wien, 1805, p. 93.

6 Curtze, loc. cit., gives the text.

0 Curtze, loc. cit., found some forty-five copies of the Algoris-
mus in three libraries of Munich, Venice, and Erfurt (Amploniana).
Examination of two manuscripts from the Plimpton collection and
the Columbia library shows such marked divergence from each other
and from the text published by Curtze that the conclusion seems legiti-
mate that these were students’ lecture notes. The shorthand char-
acter of the writing further confirms this view, as it shows that they
were written largely for the personal use of the writers.

SPREAD OF THE NUMERALS TN EUROPE 135

term Arabic numerals became common. In two places
there is mention of the inventors of this system. In the
introduction it is stated that this science of reckoning
was due to a philosopher named Algus, whence the name
algorismus,1 and hi the section on numeration reference
is made to the Arabs as the inventors of this science.2
While some of the commentators, Petrus de Dacia3
among them, knew of the Hindu origin, most of them
undoubtedly took the text as it stood ; and so the Arabs
were credited with the invention of the system.

The first definite trace that we have of an algorism
in the French language is found in a manuscript written
about 1275.4 This interesting leaf, for the part on algo-
rism consists of a single folio, was noticed by the Abbe
Lebceuf as early as 1741, 5 and by Daunou in 1824.6 It
then seems to have been lost in the multitude of Paris
manuscripts; for although Chasles7 relates his vain search
for it, it was not rediscovered until 1882. In that year
M. Ch. Henry found it, and to his care we owe our knowl-
edge of the interesting manuscript. The work is anony-
mous and is devoted almost entirely to geometry, only

1 “Quidam philosophus edidit nomine Algus, unde et Algorismus
nuncupatur.” [Curtze, loc. cit., p. 1.]

2 “ Sinistrorsum autem scribimus in hac arte more arabico sive
iudaico, hums scientiae inventorum.” [Curtze, loc. cit., p. 7.] The
Plimpton manuscript omits the words “sive iudaico.”

8 “Non enim omnis numerus per quascumque figuras Indorum
repraesentatur, sed tantum determinatus per determinatam, ut 4 non
per 5, . . [Curtze, loc. cit., p. 25.]

4 C. Henry, “Sur les deux plus anciens trails framjais d’algorisme
et de g^om^trie,” Boncompagni Bulletino, Vol. XV, p. 40; Victor
Mortet, “Le plus ancien traits frangais d’algorisme,” loc. cit.

6 L'lZtat des sciences en France , depuis la mart du Boy Robert, arrivde
en 1031, jusqu' (i celle de Philippe le Bel, arrivde en 1314, Paris, 1741.

6 Discours sur Vdtat des lettres en France au XlIIe sibcle, Paris, 1824.

7 Aperqu historique, Paris, 1876 ed., p. 404.

136

THE HIN HU-Ali ABIC NUMERALS

two pages (one folio) relating to arithmetic. In these the
forms of the numerals are given, and a very brief statement
as to the operations, it being evident that the writer him-
self had only the slightest understanding of the subject.

Once the new system was known in France, even
thus superficially, it would be passed across the Chan-
nel to England. Higden,1 writing soon after the opening
of the fourteenth century, speaks of the French influence
at that time and for some generations preceding : 2 “ For
two hundred years children in scole, agenst the usage
and manir of all other nations beeth compelled for to
leave hire own language, and for to construe liir lessons
and hire thynges in Frensche. . . . Gentilmen children
beeth taught to speke Frensche from the tyme that they
bitli rokked in hir cradell ; and uplondissche men will
likne himself to gentylmen, and fondeth with greet besy-
nesse for to speke Frensche.”

The question is often asked, why did not these new
numerals attract more immediate attention ? Why did
they have to wait until the sixteenth century to be gen-
erally used in business and in the schools ? In reply it
may be said that in their elementary work the schools
always wait upon the demands of trade. That work which
pretends to touch the life of the people must come reason-
ably near doing so. Now the computations of business
until about 1500 did not demand the new figures, for
two reasons: First, cheap paper was not known. Paper-
making of any kind was not introduced into Europe until

1 Ranulf Higden, a native of the west of England, entered St.
Werburgh’s monastery at Chester in 1299. He was a Benedictine
monk and chronicler, and died in 1364. His Polychronicon , a history
in seven books, was printed by Caxton in 1480.

2 Trevisa’s translation, Higden having written in Latin.

SPREAD OF THE NUMERALS JN EUROPE 187

the twelfth century, and cheap paper is a product of
the nineteenth. Pencils, too, of the modern type, date
only from the sixteenth century. In the second place,
modern methods of operating, particularly of multiplying
and dividing (operations of relatively greater importance
when all measures were in compound numbers requiring
reductions at every step), were not yet invented. The
old plan required the erasing of figures after they had
served their purpose, an operation very simple with coun-
ters, since they could be removed. The new plan did
not as easily permit this. Hence we find the new numer-
als very tardily admitted to the counting-house, and not
welcomed with any enthusiasm by teachers.1

Aside from their use in the early treatises on the new
art of reckoning, the numerals appeared from time to
tune in the dating of manuscripts and upon monuments.
The oldest definitely dated European document known

1 An illustration of this feeling is seen in the writings of Prosdocimo
de’ Beldomandi (b. c. 1370-1380, d. 1428): “Inveni in quam pluribus
libris algorismi nuncupatis mores circa numeros operandi satis varios
atque diversos, qui licet boni existerent atque veri erant, tamen fasti-
diosi, turn propter ipsarum regularum multitudinem, turn propter
earum deleationes, turn etiam propter ipsarum operationum proba-
tions, utrum si bone fuerint velne. Erant et etiam isti modi interim
fastidiosi, quod si in aliquo calculo astroloico error contigisset, calcu-
latorem operationem suam a capite incipere oportebat, dato quod
error suus adliuc satis propinquus existeret; et hoc propter figuras in
sua operation deletas. Indigebat etiam calculator semper aliquo
lapide vel sibi conformi, super quo scribere atque faciliter delere
posset figuras cum quibus operabatur in calculo suo. Et quia haec
omnia satis fastidiosa atque laboriosa mihi visa sunt, disposui libellum
edere in quo omnia ista abicerentur : qui etiam algorismus sive liber
de numeris denominari poterit. Scias tamen quod in hoc libello po-
nere non intendo nisi ea quae ad calculum necessaria sunt, alia quae
in aliis libris practice arismetrice tanguntur, ad calculum non neces-
saria, propter brevitatem dimitendo.” [Quoted by A. Nagl, Zeitsclirift
furMathematikund Physik, Hist. -lit. Abth. ,Vol. XXXIV, p. 143 ; Smith,
Bara Arithmetica , p. 14, in facsimile.]

138

THE IIINDU-AKABIC NUMERALS

to contain the numerals is a Latin manuscript,1 the
Codex Vigilanus, written in the Albelda Cloister not
far from Logroho in Spain, in 976 a.d. The nine char-
acters (of gobar type), without the zero, are given as an
addition to the first chapters of the third book of the
Origines by Isidorus of Seville, in which the Roman nu-
merals are under discussion. Another Spanish copy of
the same work, of 992 a.d., contains the numerals in the
corresponding section. The writer ascribes an Indian
origin to them in the following words: “Item de figuris
arithmetic^. Sene debemus in Indos subtilissimum inge-
nium habere et ceteras gentes eis in arithmetica et geo-
metria et ceteris liberalibus disciplinis concedere. Et hoc
manifestum est in nobem figuris, quibus designant unum-
quemque gradum cuiuslibet gradus. Quarum hec sunt
forma.” The nine gobar characters follow. Some of the
abacus forms2 previously given are doubtless also of
the tenth century. The earliest Arabic documents con-
taining the numerals are two manuscripts of 874 and
888 a.d.3 They appear about a century later in a work 4
written at Shiraz in 970 a.d. There is also an early
trace of their use on a pillar recently discovered in a
church apparently destroyed as early as the tenth cen-
tury, not far from the Jeremias Monastery, in Egypt.

1 P. Ewald, loc. cit. ; Franz Steffens, Lateinische Paliiographie, pp.
xxxix-xl. We are indebted to Professor J. M. Burnam for a photo-
graph of this rare manuscript.

2 See the plate of forms on p. 88.

3 Karabacek, loc. cit., p. 5G; Ivarpinski, “Hindu Numerals in the
Fihrist,” Bibliotheca Mathematical Vol. XI (3), p. 121.

4 Woepcke, “Surune donnde historique,” etc., loc. cit., and “Essai
d’une restitution de travaux perdus d’ Apollonius sur les quantity
irrationnelles, d’apr&s des indications tirdcs d’un manuscrit arabe,
Tome XIV des Mimoires presentes par divers savants h V Academic des
sciences , Paris, 1850, note, pp. 6-14.

SPREAD OF THE NUMERALS IN EUROPE 139

A graffito in Arabic on this pillar has the date 349 A.H.,
which corresponds to 961 a.d.1 For the dating of Latin
documents the Arabic forms were used as early as the
thirteenth century.2

On the early use of these numerals in Europe the
only scientific study worthy the name is that made by Mr.
G. F. Hill of the British Museum.3 From his investiga-
tions it appears that the earliest occurrence of a date in
these numerals on a coin is found in the reign of Roger
of Sicily in 1138.4 Until recently it was thought that the
earliest such date was 1217 a.d. for an Arabic piece and
1388 for a Turkish one.6 Most of the seals and medals
containing dates that were at one time thought to be
very early have been shown by Mr. Hill to be of rela-
tively late workmanship. There are, however, in Euro-
pean manuscripts, numerous instances of the use of these
numerals before the twelfth century. Besides the exam-
ple in the Codex Vigilanus, another of the tenth century
has been found in the St. Gall MS. now in the Univer-
sity Library at Zurich, the forms differing materially from
those in the Spanish codex.

The third specimen in point of time in Mr. Hill’s list is
from a Vatican MS. of 1077. The fourth and fifth speci-
mens are from the Erlangen MS. of Boethius, of the same

1 Archeological Report of the Egypt Exploration Fund for 190S-1909,
London, 1910, p. 18.

2 There was a set of astronomical tables in Boncompagni’s library
bearing this date: “Nota quod anno dni nri ihu xpi. 1204. perfecto.”
See Narducci’s Catalogo , p. 130.

3 “On the Early use of Arabic Numerals in Europe,” read before
the Society of Antiquaries April 14, 1910, and published in Archceologia
in the same year.

4 Ibid., p. 8, n. The date is part of an Arabic inscription.

6 O. Codrington, A Manual of Musalman Numismatics , London,
1904.

140

THE HINDU- ARABIC NUMERALS

(eleventh) century, and the sixth and seventh are also
from an eleventh-century MS. of Boethius at Chartres.

Earliest Manuscript Forms

These and other early forms are given by Mr. Hill in this
table, which is reproduced with his kind permission.

This is one of more than fifty tables given in Mr.
Hill’s valuable paper, and to this monograph students

SPREAD OF THE NUMERALS IN EUROPE 141

are referred for details as to the development of number-
forms in Europe from the tenth to the sixteenth cen-
tury. It is of interest to add that he has found that
among the earliest dates of European coins or medals
in these numerals, after the Sicilian one already men-
tioned, are the following : Austria, 1484 ; Germany, 1489
(Cologne) ; Switzerland, 1424 (St. Gall) ; Netherlands,
1474 ; France, 1485 ; Italy, 1390. 1

The earliest English com dated in these numerals was
struck in 1551, 2 although there is a Scotch piece of 1539. 3
In numbering pages of a printed book these numerals
were first used in a work of Petrarch’s published at Co-
logne in 1471.4 The date is given in the following form
in the Biblia Pauperuvi ,5 a block-book of 1470, while in

another block-book which possibly goes back to c. 1430 6
the numerals appear in several illustrations, with forms
as follows :

Many printed works anterior to 1471 have pages or chap-
ters numbered by hand, but many of these numerals are

1 See Arbutlinot, The Mysteries of Chronology , London, 1900, pp. 75,
78, 98 ; F. Pichler, Repertorium der steierischen Miinzlcunde, Gratz, 1875,
where the claim is made of an Austrian coin of 1458 ; Bibliotheca
Mathematical Vol. X (2), p. 120, and Vol. XII (2), p. 120. There is a
Brabant piece of 1478 in the collection of D. E. Smith.

2 A specimen is in the British Museum. [Arbuthnot, p. 79.1

3 Ibid., p. 79.

4 Liber de Remediis utriusque fortunae Coloniae.

6 Fr. Walthern et Hans Hunting, Nordlingen.

0 Ars Memorandi, one of the oldest European block-books.

142

THE HINDU-AllABIC NUMERALS

of date much later than the printing of the work. Other
works were probably numbered directly after printing.
Thus the chapters 2, 3, 4, 5, 6 in a book of 1470 1 are
numbered as follows : Capitulem zm., . . . 5m., . . . 4111.,
. . . y, . . . vi, and followed by Roman numerals.
This appears in the body of the text, in spaces left by
the printer to be fdled in by hand. Another book2 of
1470 has pages numbered by hand with a mixture of
Roman and Hindu numerals, thus,

As to monumental inscriptions,3 there was once
thought to be a gravestone at Katharein, near Troppau,
with the date 1007, and one at Biebrich of 1299. There
is 110 doubt, however, of one at Pforzheim of 1371
and one at Ulm of 1388.4 Certain numerals on Wells
Cathedral have been assigned to the thirteenth century,
but they are undoubtedly considerably later.6

The table on page 143 will serve to supplement that

1 Eusebius Caesariensis, He prcieparatione evangelica, Y enice, J enson,
1470. The above statement holds for copies in the Astor Library and
in the Harvard University Library.

2 Erancisco de Retza, Comestorium vitiorum , Niirnberg, 1470. The
copy referred to is in the Astor Library.

3 See Mauch, “Ueber den Gebrauch arabischer Ziffern und die
Veranderungen derselben,” Anzeiger fur Kunde der deutschen Vorzeit,
1801, columns 40, 81, 110, 151, 189, 229, and 208; Calmet, Recherches
sur Vorigine des chiffres d' arithmetique, plate, loc. cit.

4 Gunther, Geschichte , p. 175, n.; Mauch, loc. cit.

5 These are given by W. R. Lethaby, from drawings by J.T. Irvine,
in the Proceedings of the Society of Antiquaries, 1900, p. 200.

6 There are some ill-tabulated forms to be found in J. Bowring,
The Decimal System , London, 1854, pp. 23, 25, and in L. A. Chassant,
Dictionnaire des abreviations latines et fran^aises . . . du moyen age,

C Z- for 426

C9. A for 147 <2^ for 202

from Mr. Hill’s work.6

SPREAD OF THE NUMERALS IN EUROPE 143

Early Manuscript Forms
12 34567890

' \ ? Y 6? 2> 9 ° Twelfth century

bi ?,% 6 T,\ % i o

• -V f ^ A ‘f 6" A- ft p p 1275.,.:,,

d1'23^'^6/\ • c. 1294 A.D.

“ I (t ^ ^ O-^«.1303.V.D.

' | *7 ^ A ‘j (PA £ 0 C. 1300 A.D.

45 ^ O c. 1442 A.D.

Paris, mdccclxvi, p. 113. The best sources we have at present,
aside from the Hill monograph, are P. Treutlein, Geschichte unserer
Zahlzeichen, Karlsruhe, 1875; Cantor’s Geschichte, V ol. I, table; M.
Prou, Manuel tie paleographic latine et frangaise , 2d ed., Paris, 1892,
p. 164; A. Cappelli, Dizionario di abbreviature latine ed italiane,
Milan, 1899. An interesting early source is found in the rare Caxton
work of 1480, The Myrrour of the World. In Chap. X is a cut with
the various numerals, the chapter beginning “The fourth scyence is
called arsmetrique.” Two of the fifteen extant copies of this work
are at present in the library of Mr. J. P. Morgan, in New York.

a From the twelfth-century manuscript on arithmetic, Curtze, loc.
cit., Abhandlungen, and Nagl, loc. cit. The forms are copied from
Plate VII in Zeitschrift fiir Mathematik und Physik , Vol. XXXIV.

b From the Regensburg chronicle. Plate containing some of these
numerals in Monumenta Germaniae historica, “Scriptores” Vol. XVII,
plate to p. 184 ; Wattenbach, Anleitung zur lateinischen Palaeographie,
Leipzig, 1886, p. 102 ; Boehmer, Pontes rerum Germanicarum, Vol. Ill,
Stuttgart, 1852, p. lxv.

c French Algorismus of 1275 ; from an unpublished photograph of
the original, in the possession of II. E. Smith. See also p. 135.

d From a manuscript of Boethius c. 1294, in Mr. Plimpton’s library.
Smith, Bar a Arithmetica, Plate I.

c Numerals in a 1303 manuscript in Sigmaringen, copied from
Wattenbach, loc. cit., p. 102.

f From a manuscript, Add. Manuscript 27,589, British Museum,
1360 a.d. The work is a computus in which the date 1360 appears,
assigned in the British Museum catalogue to the thirteenth century.

s From the copy of Sacrobosco’s Algorismus in Mr. Plimpton’s
library. Date c. 1442. ' See Smith, Para Arithmetica , p. 450.

144

THE IIINDU-ARABIC NUMERALS

— 1

IT—

6T —

- 1 JIOA |.

-UTfg*.

M —

"Hi-

—

For the sake of further com-
parison, three illustrations from
works in Mr. Plimpton’s library,
reproduced from the Hava Arith-
metical may be considered. The
first is from a Latin manuscript
on arithmetic,1 of which the orig-
inal was written at Paris hi 1424
by Rollandus, a Portuguese phy-
sician, who prepared the work at
the command of John of Lan-
caster, Duke of Bedford, at one
time Protector of England and
Regent of France, to whom the
work is dedicated. The figures
show the successive powers of 2.
The second illustration is from
Luca da Firenze’s Inprencipio
clarte dabacho ,2 c. 1475, and the
third is from an anonymous manu-
script3 of about 1500.

As to the forms of the num-
erals, fashion played a leading
part until printing was invented. This tended to fix these
forms, although in writing there is still a great variation,

' " * —

• !**u • f • C ••7* S' y • i°

as witness the French 5 and the German 7 and 9.
Even in printing there is not complete uniformity,

• X-

A-

. err

. A-

•0

■ 9

(A-

. a .

. IX •

. 1 err

.7-0-

5-4-

■72--

— rr- —

— :.g- —

■ cr

. 12--

■ib-

■10-

• XI •

■3-4-

•2-A-

1 See Eara Arithmetical pp. 446-447.

2 Ibid., pp. 469-470. 3 Ibid., pp. 477-478.

SPREAD OF THE NUMERALS IN EUROPE 145

and it is often difficult for a foreigner to distinguish
between the 3 and 5 of the French types.

As to the particular numerals, the following are some
of the forms to be found in the later manuscripts and
in the early printed books.

1. In the early printed books “one” was often i, perhaps
to save types, just as some modern typewriters use the
same character for 1 and l.1 In the manuscripts the “one ”
appears in such forms as 2

2. “Two” often appears as z in the early printed books,
12 appearing as iz.3 In the medieval manuscripts the
following forms are common : 4

1 Tlie i is used for “ one ” in the Treviso arithmetic (1478), Clichto-
veus (c. 1507 ed., where both i and j are so used), Chiarini (1481),
Sacrobosco (1488 ed.), and Tzwivel (1507 ed., where jj and jz are used
for 11 and 12). This was not universal, however, for the Algorithmus
linealis of c. 1488 has a special type for 1. In a student’s notebook of
lectures taken at the University of Wurzburg in 1GG0, in Mr. Plimpton’s
library, the ones are all in the form of i.

2 Thus the date JfcP for 1580, appears in a MS. in the Lau-
rentian library at Florence. The second and the following five char-
acters are taken from Cappelli’s Dizionario , p. 380, and are from
manuscripts of the twelfth, thirteenth, fourteenth, sixteenth, seven-
teenth, and eighteenth centuries, respectively.

8 E. g. Cliiarini’s work of 1481 ; Cliclitoveus (c. 1507).

4 The first is from an algorismus of the thirteenth century, in the
Hannover Library. [See Gerhardt, “ lleber die Entstehung und
Ausbreitung des dekadisclien Zahlensystems,” loc. cit., p. 28.] The
second character is from a French algorismus, c. 1275. [Boncom-
pagni Bulletino, Vol. XV, p. 51.] The third and the following sixteen
characters are given by Cappelli, loc. cit., and are from manuscripts
of the twelfth (1), thirteenth (2), fourteenth (7), fifteenth (3), six-
teenth (1), seventeenth (2), and eighteenth (1) centuries, respectively.

3, n-, /v*. A

146

THE IILNDU-ARABIC NUMERALS

It is evident, from the early traces, that it is merely
a cursive form for the primitive =r, just as 3 comes from
=, as in the Nana Ghat inscriptions.

3. “ Three ” usually had a special type in the first printed
books, although occasionally it appears as ^ d In the •
medieval manuscripts it varied rather less than most of
the others. The following are common forms : 1 2

3,7,

4. “Four” has changed greatly; and one of the first
tests as to the age of a manuscript on arithmetic, and
the place where it was written, is the examination
of this numeral. Until the time of printing the most
common form was’ X, although the Florentine manu-
script of Leonard of Pisa’s work has the form /jo ;3
but the manuscripts show that the Florentine arithme-
ticians and astronomers rather early began to straighten
the first of these forms up to forms like 9- 4 and 4
or <9- 5 5 more closely resembling our own. The first
printed books generally used our present form 6 with the
closed top , the open top used in writing (Jf) being

1 Thus Chiarini (1481) has Z5 for 23.

2 The first of these is from a French algorismus, c. 1275. The
second and the following eight characters are given by Cappelli,
loc. cit., and are from manuscripts of the twelfth (2), thirteenth,
fourteenth, fifteenth (3), seventeenth, and eighteenth centuries,
respectively.

8 See Nagl, loc. cit.

4 Hannover algorismus, thirteenth century.

5 See the Dagomari manuscript, in Rara Arithmetical pp. 435,
437-440.

G But in the woodcuts of the Margarita Philosophica (1503) the old
forms are used, although the new ones appear in the text. In Caxton’s
Myrrour of the World (1480) the old form is used.

SPREAD OF THE NUMERALS IN EUROPE 147

purely modern. The following tire other forms of the
four, from various manuscripts : 1

A, a, y&'T,

9* ,cc,<

5. “ Five ” also varied greatly before the time of print-
ing. The following are some of the forms : 2

6. “ Six ” has changed rather less than most of the
others. The chief variation has been in the slope of the
top, as will be seen in the following : 3

<r, IQ.C. <s-,er %

7. “ Seven,” like “ four,” has assumed its present erect
form only since the fifteenth century. In medieval
times it appeared as follows : 4

a, a,ji,

1 Cappelli, loc. cit. They are partly from manuscripts of the tenth,
twelfth, thirteenth (3), fourteenth (7), fifteenth (6), and eighteenth
centuries, respectively. Those in the third line are from Chassant’s
Dictionnaire , p. 113, without mention of dates.

2 The first is from the Hannover algorismus, thirteenth century.
The second is taken from the Rollandus manuscript, 1424. The
others in the first two lines are from Cappelli, twelfth (3), fourteenth
(5), fifteenth (13) centuries, respectively. The third line is from
Chassant, loc. cit., p. 113, no mention of dates.

3 The first of these forms is from the Hannover algorismus, thir-
teenth century. The following are from Cappelli, fourteenth (3), fif-
teenth, sixteenth (2), and eighteenth centuries, respectively.

4 The first of these is taken from the Hannover algorismus, thir-
teenth century. The following forms are from Cappelli, twelfth,

148

THE IIINDU-ARABIC NUMERALS

8. “Eight,” like “six,” has changed but little. In
medieval times there are a few variants of interest as
follows : 1

In the sixteenth century, however, there was mani- -
fested a tendency to write it C/o.2

9. “Nine” has not varied as much as most of the
others. Among the medieval forms are the following : 3

0. The shape of the zero also had a varied history.
The following are common medieval forms : 4

The explanation of the place value was a serious mat-
ter to most of the early writers. If they had been using
an abacus constructed like the Russian chotu, and had
placed this before all learners of the positional system,
there would have been little trouble. But the medieval

thirteenth, fourteenth (5), fifteenth(2), seventeenth, and eighteenth
centuries, respectively.

1 All of these are given by Cappelli, thirteenth, fourteenth, fifteenth
(2), and sixteenth centuries, respectively.

2 Smith, Kara Arithmetic a, p. 489. This is also seen in several of the
Plimpton manuscripts, as in one written at Ancona in 1084. See also
Cappelli, loc. cit.

8 French algorismus, c. 1275, for the first of these forms. Cap-
pelli, thirteenth, fourteenth, fifteenth (3), and seventeenth centuries,
respectively. The last three are taken from Byzantinische Analekten,
J. L. Heiberg, being forms of the fifteenth century, but not at all
common. 9 was the old Greek symbol for 90.

4 For the first of these the reader is referred to the forms ascribed
to Boethius, in the illustration on p. 88 ; for the second, to Radulpli
of Laon, see p. GO. The third is used occasionally in the Rollandus
(1424) manuscript, in Mr. Plimpton’s library. The remaining three
are from Cappelli, fourteenth (2) and seventeenth centuries.

©, O, <f>, 6

SPREAD OF THE NUMERALS IN EUROPE 149

line-reckoning, where the lines stood for powers of 10
and the spaces for half of such powers, did not lend
itself to this comparison. Accordingly we find such
labored explanations as the following, from The Crafte
of Nombrynge :

“ Euery of these figuris bitokens hym selfe & no more,
yf he stonde in the first place of the rewele. . . .

“ If it stonde in the secunde place of the rewle, he be-
tokens ten tymes hym selfe, as this figure 2 here 20
tokens ten tyrne hym selfe, that is twenty, for he hym
selfe betokens tweyne, & ten tymes twene is twenty.
And for he stondis on the lyf t side & in the secunde
place, he betokens ten tyme hym selfe. And so go
forth. . . .

“Nil cifra significat sed dat signare sequenti. Expone
this verse. A cifre tokens nojt, bot he makes the figure
to betoken that comes after hym more than he shuld &
he were away, as thus 10. here the figure of one tokens
ten, & yf the cifre were away & no figure byfore hym he
schuld token bot one, for than he schuld stonde in the
first place. . . .” 1

It would seem that a system that was thus used for
dating documents, corns, and monuments, would have
been generally adopted much earlier than it was, par-
ticularly in those countries north of Italy where it did
not come into general use until the sixteenth century.
This, however, has been the fate of many inventions, as
witness our neglect of logarithms and of contracted proc-
esses to-day.

As to Germany, the fifteenth century saw the rise of
the new symbolism ; the sixteenth century saw it slowly
1 Smith, An Early English Algorism.

150

THE HINDU- ARABIC NUMERALS

gain the mastery ; the seventeenth century saw it finally
conquer the system that for two thousand years had
dominated the arithmetic of business. Not a little of
the success of the new plan was due to Luther’s demand
that all learning should go into the vernacular.1

During the transition period from the Roman to the
Arabic numerals, various anomalous forms found place.
For example, we have in the fourteenth century c a for
104 ; 2 1000. 300. 80 et 4 for 1384 ;3 and in a manuscript
of the fifteenth century 12901 for 1 291. 4 In the same
century m.cccc. 811 appears for 1482, 5 while M°CCCC°50
(1450) and MCCCCXL6 (1446) are used by Theodo-
ricus Ruffi about the same time.6 To the next century
belongs the form lvojj for 1502. Even in Sfortunati’s
Nuovo lume 7 the use of ordinals is quite confused, the
propositions on a single page being numbered “tertia,”
“4,” and “V.”

Although not connected with the Arabic numerals in
any direct way, the medieval astrological numerals may
here be mentioned. These are given by several early
writers, but notably by Noviomagus (1539), 8 as follows 9:

1234 5 6 789 10

’ □ Ci’

1 Kuckuck, p. 5. 2 A. Cappelli, loc. cit., p. 372.

8 Smith, Rara Arithmetical p. 443.

4 Curtze, Petri Philomeni de Dacia etc., p. ix.

6 Cappelli, loc.cit., p. 376. 0 Curtze, loc. cit., pp. viii-ix, note.

7 Edition of 1544-1546, f. 52.

8 De numeris libri II, 1544 ed., cap. xv. Heilbronner, loc. cit., p.

730, also gives them, and compares this with other systems.

9 Noviomagus says of them : “ De quibusdam Astrologicis, sive
Chaldaicis numerorum notis. . . . Sunt & alise qusedam notas, quibus
Cbaldaei & Astrologii quemlibet numerum artificiose & argut6 descri-
bunt, scitu periucundae, quas nobis communicauit Rodolplius Paluda-

nus Nouiomagus.”

SPREAD OF THE NUMERALS IN EUROPE 151

Thus we find the numerals gradually replacing the
Roman forms all over Europe, from the time of Leo-
nardo of Pisa until the seventeenth century. But in the
Far East to-day they are quite unknown in many coun-
tries, and they still have their way to make. In many
parts of India, among the common people of Japan and
China, in Siam and generally about the Malay Peninsula,
in Tibet, and among the East India islands, the natives
still adhere to their own numeral forms. Only as West-
ern civilization is making its way into the commercial
life of the East do the numerals as used by us find place,
save as the Sanskrit forms appear in parts of India. It
is therefore with surprise that the student of mathematics
comes to realize how modem are these forms so common
in the W est, how limited is their use even at the present
time, and how slow the world has been and is in adopt-
ing such a simple device as the Hindu-Arabic numerals.

INDEX

Abbo of Fleury, 122
'Abdallah ibn al-Hasan, 92
'Abdallatlf ibn Yusuf, 93
'Abdalqadir ibn 'All al-Sakhawi, 6
Abenragel, 34

Abraham ibn Me'fr ibn Ezra, see
Rabbi ben Ezra
Abu 'All al-Hosein ibn SIna, 74
Abu ’l-Hasan, 93, 100
Abu ’l-Qasim, 92
Abu ’l-Teiyib, 97
Abu Nasr, 92
Abu Roshd, 113

Abu Sahl Dunash ibn Tamim, 65,
67

Adelhard of Bath, 5, 55, 97, 119,
123, 126

Adhemar of Chabanois, 111
Ahmed al-NasawI, 98
Ahmed ibn 'Abdallah, 9, 92
Ahmed ibn Mohammed, 94
Ahmed ibn 'Omar, 93
Aksaras, 32
Alanus ab Insulis, 124
Al-Bagdadi, 93
Al-Battani, 54
Albelda (Albaida) MS., 116
Albert, J., 62
Albert of York, 103
Al-Blrunl, 6, 41, 49, 65, 92, 93
Alcuin, 103

Alexander the Great, 76
Alexander de Villa Dei, 11, 133
Alexandria, 64, 82

Al-Fazari, 92
Alfred, 103
Algebra, etymology, 5
Algerian numerals, 68
Algorism, 97

Algorismus, 124, 126, 135
Algorismus cifra, 120
Al-Hassar, 65
'All ibn Abi Bekr, 6
'All ibn Ahmed, 93, 98
Al-Karabisi, 93

Al-KhowarazmI, 4, 9, 10, 92, 97,
98, 125, 126
Al-Kindi, 10, 92
Almagest, 54
Al-MagrebI, 93
Al-Mahalli, 6
Al-Mamun, 10, 97
Al-Mansur, 96, 97
Al-Mas'udI, 7, 92
Al-Nadlm, 9
Al-NasawI, 93, 98
Alphabetic numerals, 39, 40, 43
Al-Qasim, 92
Al-Qass, 94
Al-Sakhawi, 6
Al-Sardafi, 93
Al-Sijzl, 94
Al-Sufi, 10, 92
Ambrosoli, 118
Ankapalli, 43
Apices, 87, 117, 118
Arabs, 91-98
Arbuthnot, 141

154

THE HINDU-ARABIC NUMERALS

Archimedes, 15, 10
Arcus Pictagore, 122
Arjuna, 15
Arnold, E., 15, 102
Ars memorandi, 141
Aryabhata, 39, 43, 44
Aryan numerals, 19
Aschbach, 134
Aslimole, 134
Asoka, 19, 20, 22, 81
As-sifr, 57, 58
Astrological numerals, 150
Atharva-Veda, 48, 50, 55
Augustus, 80
Averroes, 113
Avicenna, 58, 74, 113

Babylonian numerals, 28
Babylonian zero, 51
Bacon, R., 131
Bactrian numerals, 19, 30
Bseda, 2, 72
Bagdad, 4, 96

Bakhsali manuscript, 43, 49, 52, 53

Ball, C. J., 35

Ball, W. W. R., 36, 131

Bana, 44

Barth, A., 39

Bayang inscriptions, 39

Bayer, 33

Bayley, E. C., 19, 23, 30, 32, 52, 89

Beazley, 75

Bede, see Bseda

Beldomandi, 137

Beloch, J., 77

Bendall, 25, 52

Benfey, T., 26

Bernelinus, 88, 112, 117, 121

Besagne, 128

Besant, W., 109

Bettino, 36

Bhandarkar, 18, 47, 50

Bhaskara, 53, 55
Biernatzki, 32
Biot, 32

Bjornbo, A. A., 125, 126
Blassi^re, 119
Bloomfield, 48
Blume, 85
Boeckh, 62
Boehmer, 143
Boeschenstein, 119
Boethius, 63, 70-73, 83-90
Boissiere, 63
Bombelli, 81
Bonaini, 128

Boncompagni, 5, 6, 10, 48, 50, 123,
125

Borghi, 59
Borgo, 119
Bougie, 130
Bowring, J., 56
Brahmagupta, 52
Bralnnanas, 12, 13
Brahml, 19, 20, 31, 83
Brandis, J., 54
Brhat-Samhita, 39, 44, 78
Brockhaus, 43
Bubnov, 65, 84, 110, 116
Buddha, education of, 15, 16
Biidinger, 110
Bugia, 130

Bidder, G., 15, 19, 22, 31, 44, 50
Burgess, 25
Burk, 13

Burmese numerals, 36
Burnell, A. C., 18, 40
Buteo, 61

Calandri, 59, 81
Caldwell, R., 19
Calendars, 133
Calmet, 34

Cantor, M., 5, 13, 30, 43, 84

INDEX

155

Capella, 86
Cappelli, 143

Caracteres, 87, 113, 117, 119
Cardan, 119

Carmen de Algorismo, 11, 134
Casagrandi, 132
Casiri, 8, 10
Cassiodorus, 72
Cataldi, 62
Cataneo, 3
Caxton, 143, 146
Ceretti, 32
Ceylon numerals, 36
Chalfont, F. II., 28
Champenois, 60
Characters, see Caracteres
Charlemagne, 103
Chasles, 54, 60, 85, 116, 122,
135

Chassant, L. A., 142
Chaucer, 121
Chiarini, 145, 146
Cliiffre, 58

Chinese numerals, 28, 56
Chinese zero, 56
Cifra, 120, 124
Cipher, 58
Circulus, 58, 60
Clichtoveus, 61, 119, 145
Codex Vigilanus, 138
Codrington, O., 139
Coins dated, 141
Colebrooke, 8, 26, 46, 53
Constantine, 104, 105
Cosmas, 82
Cossali, 5
Counters, 117
Courteille, 8
Coxe, 59

Crafte of Nombrynge, 11, 87, 149
Crusades, 109
Cunningham, A., 30, 75

Curtze, 55, 59, 126, 134
Cyfra, 55

Dagomari, 146
D’Alviella, 15
Dante, 72

Dasypodius, 33, 57, 63
Daunou, 135
Delambre, 54
Devanagari, 7
Devoulx, A., 68
Dhruva, 49, 50
Dicsearchus of Messana, 77
Digits, 119
Diodorus Siculus, 76
Du Cange, 62
Dumesnil, 36

Dutt, R. C., 12, 15, 18, 75
DvivedI, 44

East and West, relations, 73-81,
100-109

Egyptian numerals, 27
Eisenlohr, 28
Elia Misrachi, 57
Enchiridion Algorismi, 58
Enestrom, 5, 48, 59, 97, 125, 128
Europe, numerals in, 63, 99, 128,
136

Eusebius Caesariensis, 142
Euting, 21
Ewald, P., 116

Fazzari, 53, 54

Fibonacci, see Leonardo of Pisa

Figura nihili, 58

Figures, J.19. See numerals.

Fihrist, 67, 68, 93

Finaeus, 57

FirdusI, 81

Fitz Stephen, W., 109
Fleet, J. C., 19, 20, 50

156

THE HINDU- ARABIC NUMERALS

Floras, 80
Fliigel, G., 68
Francisco de Retza, 142
FraiiQois, 58

Friedlein, G., 84, 113, 116, 122
Froude, J. A., 129

Gandhara, 19
Garbe, 48
Gasbarri, 58

Gautier de Coincy, 120, 124
Gemma Frisius, 2, 3, 119
Gerber, 113

Gerbert, 108, 110-120, 122

Gerhardt, C. I., 43, 56, 93, 118

Gerland, 88, 123

Gherard of Cremona, 125

Gibbon, 72

Giles, H. A., 79

Ginanni, 81

Giovanni di Danti, 58

Glareanus, 4, 119

Gnecchi, 71, 117

Gobar numerals, 65, 100, 112,
124, 138
Gow, J., 81
Grammateus, 61
Greek origin, 33
Green, J. R., 109
Greenwood, I., 62, 119
Guglielmini, 128
Gulistan, 102
Gunther, S., 131
Guyard, S., 82

Habash, 9, 92
Hager, J. (G.), 28, 32
Halliwell, 59, 85
Hankel, 93

Ilarun al-Rashid, 97, 106
Ilavet, 110
Heath, T. L., 125

Hebrew numerals, 127
Hecatseus, 75
Heiberg, J. L., 55, 85, 148
Heilbronner, 5

Henry, C., 5, 31, 55, 87, 120,
135

Heriger, 122

Hermannus Contractus, 123
Herodotus, 76, 78
Ileyd, 75
Higden, 136
Hill, G. F., 52, 139, 142
Hillebrandt, A., 15, 74
Ililprecht, II. V., 28
Hindu forms, early, 12
Hindu number names, 42
Hodder, 62
Hoernle, 43, 49
Ilolywood, see Sacrobosco
Hopkins, E. W., 12
Horace, 79, 80

Hosein ibn Mohammed al-Ma-
lialli, 6

Hostus, M., 56
Howard, H. II., 29
Ilrabanus Maurus, 72
Huart, 7
Huet, 33
Hugo, H., 57
Humboldt, A. von, 62
Huswirt, 58

Iamblichus, 81
Ibn Abi Ya'qub, 9
Ibn al-Adami, 92
Ibn al-Banna, 93
Ibn Khordadbeh, 101, 106
Ibn Wahab, 103
India, history of, 14
writing in, 18
Indicopleustes, 83
Indo-Bactrian numerals, 19

INDEX

157

IndrajI, 23

Ishaq ibn Yusuf al-Sardafl, 93

Jacob of Florence, 57
Jacquet, E., 38
Jamshid, 56
Jehan Certain, 59
Jetons, 58, 117
Jevons, F. B., 76
Johannes Ilispalensis, 48, 88, 124
John of Halifax, see Sacrobosco
John of Luna, see Johannes Ilis-
palensis

Jordan, L., 58, 124
Joseph Ispanus (Joseph Sapiens),
115

Justinian, 104

Kale, M. K., 26
Karabacek, 56

Karpinski, L. C., 126, 134, 138
Katyayana, 39

Kaye, C. R., 6, 16, 43, 46, 121

Keane, J., 75, 82

Keene, H. G., 15

Kern, 44

KharosthI, 19, 20

Khosru, 82, 91

Kielhorn, F., 46, 47

Kirch er, A., 34

Ivitab al-Fihrist, see Fihrist

Kleinwachter, 32

Klos, 62

Kobel, 4, 58, 60, 119, 123
Krumbacher, K., 57
Kuckuck, 62, 133
Kugler, F. X., 51

Lachmann, 85
Lacouperie, 33, 35
Lalitavistara, 15, 17
Laini, G., 57

La Roche, 61
Lassen, 39
Latyayana, 39
Lebceuf, 135

Leonardo of Pisa, 5, 10, 57, 64,
74, 120, 128-133
Lethaby, W. R., 142
Levi, B., 13
Levias, 3
Libri, 73, 85, 95
Light of Asia, 16
Luca da Firenze, 144
Lucas, 128

Maliabharata, 18
Mahavlracarya, 53
Malabar numerals, 36
Malayalam numerals, 36
Mannert, 81

Margarita Pliilosophica, 146
Marie, 78
Marquardt, J., 85
Marshman, J. C., 17
Martin, T. II., 30, 62, 85, 113
Martines, D. C., 58
Mashallah, 3
Maspero, 28
Maucli, 142

MaximusPlanudes,2,57,66,93,120

Megasthenes, 77

Merchants, 114

Meynard, 8

Migne, 87

Mikami, Y., 56

Milanesi, 128

Mohammed ibn 'Abdallah, 92
Mohammed ibn Ahmed, 6
Mohammed ibn 'All 'Abdl, 8
Mohammed ibn Musa, see Al-
Khowarazmi
Molinier, 123
Monier-Williams, 17

158

THE IIINDU-AEABIC NUMERALS

Morley, D., 126
Moroccan numerals, 68, 119
Mortet, V., 11
Moseley, C. B., 33
Motalihar ibn Tahir, 7
Mueller, A., 68
Mumforcl, J. K., 109
Muwaffaq al-DIn, 93

Nabatean forms, 21
Nallino, 4, 54, 55
Nagl, A., 55, 110, 113, 126
Nana Gliat inscriptions, 20, 22,
23, 40

Narducci, 123
Nasilc cave inscriptions, 24
Nazif ibn Yumn, 94
Neander, A., 75
Neophytos, 57, 62
Neo-Pythagoreans, 64
Nesselmann, 53
Newman, Cardinal, 96
Newman, F. W., 131
Noldeke, Th., 91
Notation, 61
Note, 61, 119

Noviomagus, 45, 61, 119, 150

Nidi, 61

Numerals,

Algerian, 68
astrological, 150
Brahmi, 19-22, 83
early ideas of origin, 1
Hindu, 26

Hindu, classified, 19, 38
KharosthI, 19-22
Moroccan, 68
Nabatean, 21
origin, 27, 30, 31, 37
supposed Arabic origin, 2
supposed Babylonian origin,
28

Numerals,

supposed Chaldean and Jew-
ish origin, 3

supposed Chinese origin, 28,
32

supposed Egyptian origin, 27,
30, 69, 70

supposed Greek origin, 33
supposedPlioenician origin, 32
tables of, 22-27, 36, 48, 50,
69, 88, 140, 143, 145-148

O’Creat, 5, 55, 119, 120
Olleris, 110, 113
Oppert, G., 14, 75

Pali, 22

Pancasiddhantika, 44
Paravey, 32, 57
Patallputra, 77
Patna, 77
Patrick, R., 119
Payne, E. J., 106
Pegolotti, 107
Peletier, 2, 62
Perrot, 80
Persia, 66, 91, 107
Pertz, 115

Petrus de Dacia, 59, 61, 62
Pez, P. B., 117
“Pliilalethes,” 75
Phillips, G,, 107
'Picavet, 105
Pichler, F., 141
Pilian, A. P., 36
Pisa, 128

Place value, 26, 42, 46, 48
Planudes, see Maximus Planudes
Plimpton, G. A., 56, 59, 85, 143,
144, 145, 148
Pliny, 76

Polo, N. and M., 107

INDEX

159

Prandel, J. G., 54
Prinsep, J., 20, 31
Propertius, 80

Prosdocimo de’ Beldomandi, 137

Prou, 143

Ptolemy, 54, 78

Putnam, 103

Pythagoras, 63

Pythagorean numbers, 13

Pytheas of Massilia, 76

Rabbi ben Ezra, 60, 127
Radulph of Laon, 60, 113, 118, 124
Raets, 62

Rainer, see Gemma Erisius
Ramayana, 18
Ramus, 2, 41, 60, 61
Raoul Glaber, 123
Rapson, 77

Rauhfuss, see Dasypodius
Raumer, K. von, 111
Reclus, E., 14, 96, 130
Recorde, 3, 58
Reinaud, 67, 74, 80
Reveillaud, 36
Richer, 110, 112, 115
Riese, A., 119
Robertson, 81

Robertas Cestrensis, 97, 126

Rodet, 5, 44

Roediger, J., 68

Rollandus, 144

Romagnosi, 81

Rosen, F., 5

Rotula, 60

Rudolff, 85

Rudolph, 62, 67

Ruffi, 150

Sachau, 6

Sacrobosco, 3, 58, 133
Sacy, S. de, 66, 70

Sa'di, 102

Saka inscriptions, 20
Samu’Il ibn Yahya, 93
Sarada characters, 55
Savonne, 60
Scaliger, J. C., 73
Scheubel, 62
Schlegel, 12
Schmidt, 133
Schonerus, 87, 119
Schroeder, L. von, 13
Scylax, 75
Sedillot, 8, 34
Senart, 20, 24, 25
Sened ibn 'All, 10, 98
Sfortunati, 62, 150
Shelley, W., 126
Siamese numerals, 36
Siddhanta, 8, 18
Sifr, 57
Sigsboto, 55
Sihab al-DIn, 67
Silberberg, 60
Simon, 13

Sinan ibn al-Fath, 93
Sindbad, 100
Sindhind, 97
Sipos, 60
Sirr, II. C., 75
Skeel, C. A., 74

Smith, D.E., 11, 17, 53,86, 141, 143

Smith, V. A., 20, 35, 46, 47

Smith, Win., 75

Smrti, 17

Spain, 64, 65, 100

Spitta-Bey, 5

Sprenger, 94

Srautasutra, 39

Steffens, F., 116

Steinschneider, 5, 57, 65, 66, 98,
126

Stifel, 62

160

THE HINDU- ARABIC NUMERALS

Subandhus, 44

Suetonius, 80

Suleiman, 100

Sunya, 43, 53, 57

Suter, 5, 9, 68, 69, 93, 116, 131

Sutras, 13

Sykes, P. M., 75

Sylvester II, see Gerbert

Symonds, J. A., 129

Tannery, P., 62, 84, 85

Tartaglia, 4, 61

Taylor, I., 19, 30

Teca, 55, 61

Tennent, J. E., 75

Texada, 60

Theca, 58, 61

Theophanes, 64

Thibaut, G., 12, 13, 16, 44, 47

Tibetan numerals, 36

Timotlieus, 103

Tonstall, C., 3, 61

Trenchant, 60

Treutlein, 5, 63, 123

Trevisa, 136

Treviso arithmetic, 145

Trivium and quadrivium, 73

Tsin, 56

Tunis, 65

Turchill, 88, 118, 123
Tumour, G., 75
Tziphra, 57, 62
Tflcppa., 55, 57, 62
Tzwivel, 61, 118, 145

Ujjain, 32
Unger, 133
Upanishads, 12
Usk, 121

Valla, G., 61
Van der Schuere, 62

Varaha-Mihira, 39, 44, 78
Vasavadatta, 44
Vaux, Carra de, 9, 74
Vaux, W. S. W., 91
Vedaiigas, 17
Vedas, 12, 15, 17
Vergil, 80

Vincent, A. J. II., 57
Vogt, 13
Voizot, P., 36
Vossius, 4, 76, 81, 84

Wallis, 3, 62, 84, 116
Wappler, E., 54, 126
Waschke, H., 2, 93
Wattenbach, 143
Weber, A., 31
Weidler, I. F., 34, 66
Weidler, I. E. and G. I., 63, 66
Weissenborn, 85, 110
Wertheim, G., 57, 61
Whitney, W. D.. 13
Wilford, F., 75
Wilkens, 62
Wilkinson, J. G., 70
Willichius, 3

Woepcke, 3, 6, 42, 63, 64, 65, 67,
69, 70, 94, 113, 138
Wolack, G., 54
Woodruff, C. E., 32
Word and letter numerals, 38,
44

Wiistenfeld, 74

Yule, II., 107

Zepliiram, 57, 58
Zephyr, 59
Zepiro, 58

Zero, 26, 38, 40, 43, 45, 50, 51-62,
67

Zeuero, 58

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